# On certain functional equation in prime rings

On certain functional equation in prime rings IntroductionThroughout, RRwill represent an associative ring with center Z(R)Z\left(R). Given an integer n>1n\gt 1, a ring RRis said to be nn-torsion free, if for x∈Rx\in R, nx=0nx=0implies x=0x=0. The commutator xy−yxxy-yxwill be denoted by [x,y]{[}x,y]. A ring RRis prime if for a,b∈Ra,b\in R, aRb=(0)aRb=\left(0)implies that either a=0a=0or b=0b=0and is semiprime in case aRa=(0)aRa=\left(0)implies a=0a=0. We denote by Qmr,Qs,{Q}_{mr},{Q}_{s},and CCthe maximal Martindale right ring of quotients, symmetric Martindale ring of quotients, and extended centroid of a semiprime ring RR, respectively (see [1], Chapter 2). An additive mapping D:R→RD:R\to Ris called a derivation if D(xy)=D(x)y+xD(y)D\left(xy)=D\left(x)y+xD(y)holds for all pairs x,y∈Rx,y\in Rand is called a Jordan derivation in case D(x2)=D(x)x+xD(x)D\left({x}^{2})=D\left(x)x+xD\left(x)is fulfilled for all x∈Rx\in R. A derivation D:R→RD:R\to Ris inner in case DDis of the form D(x)=[a,x]D\left(x)={[}a,x]for all x∈Rx\in Rand some fixed a∈Ra\in R. Every derivation is Jordan derivation. The converse is in general not true. A classical result of Herstein [2] asserts that any Jordan derivation on a prime ring with characteristic different from two is a derivation. A brief proof of Herstein theorem can be found in [3]. In [4], one can find a generalization of Herstein theorem. Cusack [5] generalized Herstein theorem to 2-torsion free semiprime rings (see [6] for an alternative proof). Herstein theorem has been fairly generalized by Beidar et al. [7]. For results related to Herstein theorem, we refer to [8,9, 10,11]. We proceed with the following result proved by Brešar [12] (see [13] for a generalization).Theorem 1Let R be a 2-torsion free semiprime ring and let D:R→RD:R\to Rbe an additive mapping satisfying the relation.(1)D(xyx)=D(x)yx+xD(y)x+xyD(x)D\left(xyx)=D\left(x)yx+xD(y)x+xyD\left(x)for all pairs x,y∈Rx,y\in R. In this case, DDis a derivation.An additive mapping satisfying the relation (1) on an arbitrary ring is called a Jordan triple derivation. It is easy to prove that any Jordan derivation on a 2-torsion free ring is a Jordan triple derivation, which means that Theorem 1 generalizes Cusack’s generalization of Herstein theorem.Motivated by Theorem 1, Vukman et al. [14] have proved the following result (see [15] for a generalization).Theorem 2Let R be a 2-torsion free semiprime ring and let F:R→RF:R\to Rbe an additive mapping satisfying the relation(2)T(xyx)=T(x)yx−xT(y)x+xyT(x)T\left(xyx)=T\left(x)yx-xT(y)x+xyT\left(x)for all pairs x,y∈Rx,y\in R. In this case, FFis of the form 2T(x)=qx+xq2T\left(x)=qx+xq, where q∈Qs(R)q\in {Q}_{s}\left(R)is some fixed element.We proceed with the following functional equation: (3)F(xyx)=F(xy)x−xF(y)x+xF(yx),F\left(xyx)=F\left(xy)x-xF(y)x+xF(yx),which appears naturally in the proof of Theorem 2 in [16]. One can easily prove that in case we have an additive mapping F:R→R,F:R\to R,where RRis 2-torsion free semiprime ring, satisfying the relation (3) for all pairs x,y∈Rx,y\in R, then FFis of the form 2F(x)=D(x)+ax+xa,2F\left(x)=D\left(x)+ax+xa,where D:R→RD:R\to Ris a derivation and a∈Ra\in Rsome fixed element (see [16] for the details). In [16], one can find the following conjecture.Conjecture 3Let R be a 2-torsion free semiprime ring and let F:R→RF:R\to Rbe an additive mapping satisfying the relation (3) for all pairs x,y∈Rx,y\in R. In this case, F is of the form 2F(x)=D(x)+qx+xq2F\left(x)=D\left(x)+qx+xqfor all x∈Rx\in R, where D:R→RD:R\to Ris a derivation and q∈Qs(R)q\in {Q}_{s}\left(R)some fixed element.By our knowledge, the aforementioned conjecture is still an open question. The substitution y=xy=xin (1), (2), and (3) gives (4)D(x3)=D(x)x2+xD(x)x+x2D(x),\hspace{0.45em}D\left({x}^{3})=D\left(x){x}^{2}+xD\left(x)x+{x}^{2}D\left(x),(5)F(x3)=F(x)x2−xF(x)x+x2F(x)F\left({x}^{3})=F\left(x){x}^{2}-xF\left(x)x+{x}^{2}F\left(x)and (6)F(x3)=F(x2)x−xF(x)x+xF(x2).F\left({x}^{3})=F\left({x}^{2})x-xF\left(x)x+xF\left({x}^{2}).The relation (4) has been considered in [7] (actually, much more general situation has been considered). A result related to (5) can be found in [15]. It is our aim in this paper to prove the following result, which is related to the aforementioned conjecture.Theorem 4Let RRbe a prime ring of characteristic different from two and three, and let F:R→RF:R\to Rbe an additive mapping satisfying the relation(7)F(x3)=F(x2)x−xF(x)x+xF(x2)F\left({x}^{3})=F\left({x}^{2})x-xF\left(x)x+xF\left({x}^{2})for all x∈Rx\in R. In this case, F is of the form 4F(x)=D(x)+qx+xq4F\left(x)=D\left(x)+qx+xq, where D:R→RD:R\to Ris a derivation, and q∈Qs(R)q\in {Q}_{s}\left(R)is some fixed element.Main resultsAs the main tool in this paper, we use the theory of functional identities (Brešar-Beidar-Chebotar theory). The theory of functional identities considers set-theoretic maps on rings that satisfy some identical relations. When treating such relations, one usually concludes that the form of the mappings involved can be described, unless the ring is very special. We refer the reader to [17] for the introductory account on the theory of functional identities, where Brešar presents this theory and its applications to a wider audience and to [18] for the full treatment of this theory.Let RRbe an algebra over a commutative ring ϕ\phi and let (8)p(x1,x2,x3)=∑π∈S3xπ(1)xπ(2)xπ(3)p\left({x}_{1},{x}_{2},{x}_{3})=\sum _{\pi \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}{x}_{\pi \left(2)}{x}_{\pi \left(3)}be a fixed multilinear polynomial in noncommuting indeterminates xi{x}_{i}over ϕ\phi . Here, S3{{\mathbb{S}}}_{3}stands for the symmetric group of order 3. Let ℒ{\mathcal{ {\mathcal L} }}be a subset of RRclosed under pp, i.e., p(x¯3)∈ℒp\left({\bar{x}}_{3})\in {\mathcal{ {\mathcal L} }}for all x1,x2,x3∈ℒ{x}_{1},{x}_{2},{x}_{3}\in {\mathcal{ {\mathcal L} }}, where x¯3=(x1,x2,x3){\bar{x}}_{3}=\left({x}_{1},{x}_{2},{x}_{3}). We shall consider a mapping D:ℒ→RD:{\mathcal{ {\mathcal L} }}\to Rsatisfying (9)F(p(x¯3))=∑π∈S3(F(xπ(1)xπ(2))xπ(3)−xπ(1)F(xπ(2))xπ(3)+xπ(1)F(xπ(2)xπ(3)))F\left(p\left({\bar{x}}_{3}))=\sum _{\pi \in {{\mathbb{S}}}_{3}}\left(F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){x}_{\pi \left(3)}-{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}+{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)}))for all x1,x2,x3∈ℒ{x}_{1},{x}_{2},{x}_{3}\in {\mathcal{ {\mathcal L} }}. Let us mention that the idea of considering the expression [p(x¯3),p(y¯3)]\left[p\left({\overline{x}}_{3}),p({\overline{y}}_{3})]in its proof is taken from [19]. For the proof of Theorem 4, we need Theorem 5, which might be of independent interest.Theorem 5Let ℒ{\mathcal{ {\mathcal L} }}be a 6-free Lie subring of R closed under p. If T:ℒ→RT:{\mathcal{ {\mathcal L} }}\to Ris an additive mapping satisfying (9), then there exists q∈Rq\in Rsuch that 4F(x)=D(x)+xq+qx4F\left(x)=D\left(x)+xq+qxfor all x∈ℒx\in {\mathcal{ {\mathcal L} }}.ProofFor any a∈Ra\in Rand x¯3∈ℒ3{\bar{x}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}, we have [p(x¯3),a]=p([x1,a],x2,x3)+p(x1,[x2,a],x3)+p(x1,x2,[x3,a]).\left[p\left({\bar{x}}_{3}),a]=p\left(\left[{x}_{1},a],{x}_{2},{x}_{3})+p\left({x}_{1},\left[{x}_{2},a],{x}_{3})+p\left({x}_{1},{x}_{2},\left[{x}_{3},a]).Thus, (10)F[p(x¯3),a]=F(p([x1,a],x2,x3))+F(p(x1,[x2,a],x3))+F(p(x1,x2,[x3,a])).F\left[p\left({\bar{x}}_{3}),a]=F\left(p\left(\left[{x}_{1},a],{x}_{2},{x}_{3}))+F\left(p\left({x}_{1},\left[{x}_{2},a],{x}_{3}))+F\left(p\left({x}_{1},{x}_{2},\left[{x}_{3},a])).By using (10), it follows that F[p(x¯3),a]=∑π∈S3F([xπ(1),a]xπ(2))xπ(3)−∑π∈S3[xπ(1),a]F(xπ(2))xπ(3)+∑π∈S3[xπ(1),a]F(xπ(2)xπ(3))+∑π∈S3F(xπ(1)[xπ(2),a])xπ(3)−∑π∈S3xπ(1)F([xπ(2),a])xπ(3)+∑π∈S3xπ(1)F([xπ(2),a]xπ(3))+∑π∈S3F(xπ(1)xπ(2))[xπ(3),a]−∑π∈S3xπ(1)F(xπ(2))[xπ(3),a]+∑π∈S3xπ(1)F(xπ(2)[xπ(3),a])=∑π∈S3F([xπ(1)xπ(2),a])xπ(3)−∑π∈S3[xπ(1),a]F(xπ(2))xπ(3)+∑π∈S3[xπ(1),a]F(xπ(2)xπ(3))−∑π∈S3xπ(1)F([xπ(2),a])xπ(3)+∑π∈S3xπ(1)F([xπ(2)xπ(3),a])+∑π∈S3F(xπ(1)xπ(2))[xπ(3),a]−∑π∈S3xπ(1)F(xπ(2))[xπ(3),a].\begin{array}{rcl}F\left[p\left({\bar{x}}_{3}),a]& =& \displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}F\left(\left[{x}_{\pi \left(1)},a]{x}_{\pi \left(2)}){x}_{\pi \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)},a]F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)},a]F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)})\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}F\left({x}_{\pi \left(1)}\left[{x}_{\pi \left(2)},a]){x}_{\pi \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)},a]){x}_{\pi \left(3)}+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)},a]{x}_{\pi \left(3)})\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},a]-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},a]+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}\left[{x}_{\pi \left(3)},a])\\ & =& \displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},a]){x}_{\pi \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)},a]F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)},a]F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)})\\ & & -\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)},a]){x}_{\pi \left(3)}+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},a])+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},a]\\ & & -\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},a].\end{array}In particular, (11)F[p(x¯3),p(y¯3)]=∑π∈S3F([xπ(1)xπ(2),p(y¯3)])xπ(3)−∑π∈S3[xπ(1),p(y¯3)]F(xπ(2))xπ(3)+∑π∈S3[xπ(1),p(y¯3)]F(xπ(2)xπ(3))−∑π∈S3xπ(1)F([xπ(2),p(y¯3)])xπ(3)+∑π∈S3xπ(1)F([xπ(2)xπ(3),p(y¯3)])+∑π∈S3F(xπ(1)xπ(2))[xπ(3),p(y¯3)]−∑π∈S3xπ(1)F(xπ(2))[xπ(3),p(y¯3)]\hspace{-38.6em}\begin{array}{rcl}F\left[p\left({\bar{x}}_{3}),p({\bar{y}}_{3})]& =& \displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},p({\bar{y}}_{3})]){x}_{\pi \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)},p({\bar{y}}_{3})]F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)},p({\bar{y}}_{3})]F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)})-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)},p({\bar{y}}_{3})]){x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},p({\bar{y}}_{3})])+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},p({\bar{y}}_{3})]\\ & & -\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},p({\bar{y}}_{3})]\end{array}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}. For i=1,2i=1,2, we have (12)F[xπ(i)xπ(i+1),p(y¯3)]=−F[p(y¯3),xπ(i)xπ(i+1)]=∑σ∈S3F([xπ(i)xπ(i+1),yσ(1)yσ(2)])yσ(3)−∑σ∈S3[xπ(i)xπ(i+1),yσ(1)]F(yσ(2))yσ(3)+∑σ∈S3[xπ(i)xπ(i+1),yσ(1)]F(yσ(2)yσ(3))−∑σ∈S3yσ(1)F([xπ(i)xπ(i+1),yσ(2)])yσ(3)\hspace{-38.7em}\begin{array}{rcl}F\left[{x}_{\pi \left(i)}{x}_{\pi \left(i+1)},p({\bar{y}}_{3})]& =& -F\left[p({\bar{y}}_{3}),{x}_{\pi \left(i)}{x}_{\pi \left(i+1)}]\\ & =& \displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F\left(\left[{x}_{\pi \left(i)}{x}_{\pi \left(i+1)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}-\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(i)}{x}_{\pi \left(i+1)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(i)}{x}_{\pi \left(i+1)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)})-\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(i)}{x}_{\pi \left(i+1)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}\end{array}(12)+∑σ∈S3yσ(1)F([xπ(i)xπ(i+1),yσ(2)yσ(3)])+∑σ∈S3F(yσ(1)yσ(2))[xπ(i)xπ(i+1),yσ(3)]−∑σ∈S3yσ(1)F(yσ(2))[xπ(i)xπ(i+1),yσ(3)]\begin{array}{rcl}& & +\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(i)}{x}_{\pi \left(i+1)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}])+\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)})\left[{x}_{\pi \left(i)}{x}_{\pi \left(i+1)},{y}_{\sigma \left(3)}]\\ & & -\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)})\left[{x}_{\pi \left(i)}{x}_{\pi \left(i+1)},{y}_{\sigma \left(3)}]\end{array}and F[xπ(2),p(y¯3)]=−F[p(y¯3),xπ(2)]=∑σ∈S3F([xπ(2),yσ(1)yσ(2)])yσ(3)−∑σ∈S3[xπ(2),yσ(1)]F(yσ(2))yσ(3)+∑σ∈S3[xπ(2),yσ(1)]F(yσ(2)yσ(3))−∑σ∈S3yσ(1)F([xπ(2),yσ(2)])yσ(3)+∑σ∈S3yσ(1)F([xπ(2),yσ(2)yσ(3)])+∑σ∈S3F(yσ(1)yσ(2))[xπ(2),yσ(3)]−∑σ∈S3yσ(1)F(yσ(2))[xπ(2),yσ(3)]\hspace{-40.65em}\begin{array}{rcl}F\left[{x}_{\pi \left(2)},p({\bar{y}}_{3})]& =& -F\left[p({\bar{y}}_{3}),{x}_{\pi \left(2)}]\\ & =& \displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}-\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)})-\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}])+\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)})\left[{x}_{\pi \left(2)},{y}_{\sigma \left(3)}]\\ & & -\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)})\left[{x}_{\pi \left(2)},{y}_{\sigma \left(3)}]\end{array}for all y¯3∈ℒ3{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}. Therefore, (11) can be written as follows: F[p(x¯3),p(y¯3)]=∑π∈S3∑σ∈S3F([xπ(1)xπ(2),yσ(1)yσ(2)])yσ(3)xπ(3)−∑π∈S3∑σ∈S3[xπ(1)xπ(2),yσ(1)]F(yσ(2))yσ(3)xπ(3)+∑π∈S3∑σ∈S3[xπ(1)xπ(2),yσ(1)]F(yσ(2)yσ(3))xπ(3)−∑π∈S3∑σ∈S3yσ(1)F([xπ(1)xπ(2),yσ(2)])yσ(3)xπ(3)+∑π∈S3∑σ∈S3yσ(1)F([xπ(1)xπ(2),yσ(2)yσ(3)])xπ(3)+∑π∈S3∑σ∈S3F(yσ(1)yσ(2))[xπ(1)xπ(2),yσ(3)]xπ(3)−∑π∈S3∑σ∈S3yσ(1)F(yσ(2))[xπ(1)xπ(2),yσ(3)]xπ(3)−∑π∈S3∑σ∈S3[xπ(1),yσ(1)yσ(2)yσ(3)]F(xπ(2))xπ(3)+∑π∈S3∑σ∈S3[xπ(1),yσ(1)yσ(2)yσ(3)]F(xπ(2)xπ(3))−∑π∈S3∑σ∈S3xπ(1)F([xπ(2),yσ(1)yσ(2)])yσ(3)xπ(3)+∑π∈S3∑σ∈S3xπ(1)[xπ(2),yσ(1)]F(yσ(2))yσ(3)xπ(3)−∑π∈S3∑σ∈S3xπ(1)[xπ(2),yσ(1)]F(yσ(2)yσ(3))xπ(3)+∑π∈S3∑σ∈S3xπ(1)yσ(1)F([xπ(2),yσ(2)])yσ(3)xπ(3)−∑π∈S3∑σ∈S3xπ(1)yσ(1)F([xπ(2),yσ(2)yσ(3)])xπ(3)−∑π∈S3∑σ∈S3xπ(1)F(yσ(1)yσ(2))[xπ(2),yσ(3)]xπ(3)+∑π∈S3∑σ∈S3xπ(1)yσ(1)F(yσ(2))[xπ(2),yσ(3)]xπ(3)+∑π∈S3∑σ∈S3xπ(1)F([xπ(2)xπ(3),yσ(1)yσ(2)])yσ(3)−∑π∈S3∑σ∈S3xπ(1)[xπ(2)xπ(3),yσ(1)]F(yσ(2))yσ(3)+∑π∈S3∑σ∈S3xπ(1)[xπ(2)xπ(3),yσ(1)]F(yσ(2)yσ(3))−∑π∈S3∑σ∈S3xπ(1)yσ(1)F([xπ(2)xπ(3),yσ(2)])yσ(3)+∑π∈S3∑σ∈S3xπ(1)yσ(1)F([xπ(2)xπ(3),yσ(2)yσ(3)])+∑π∈S3∑σ∈S3xπ(1)F(yσ(1)yσ(2))[xπ(2)xπ(3),yσ(3)]−∑π∈S3∑σ∈S3xπ(1)yσ(1)F(yσ(2))[xπ(2)xπ(3),yσ(3)]+∑π∈S3∑σ∈S3F(xπ(1)xπ(2))[xπ(3),yσ(1)yσ(2)yσ(3)]−∑π∈S3∑σ∈S3xπ(1)F(xπ(2))[xπ(3),yσ(1)yσ(2)yσ(3)]\begin{array}{rcl}F\left[p\left({\bar{x}}_{3}),p({\bar{y}}_{3})]& =& \displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)}){x}_{\pi \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]){x}_{\pi \left(3)}+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)})\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(3)}]{x}_{\pi \left(3)}\\ & & -\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)})\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(3)}]{x}_{\pi \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)})-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)}){x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]){x}_{\pi \left(3)}\\ & & -\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)})\left[{x}_{\pi \left(2)},{y}_{\sigma \left(3)}]{x}_{\pi \left(3)}+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)})\left[{x}_{\pi \left(2)},{y}_{\sigma \left(3)}]{x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)})-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}])+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)})\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(3)}]\\ & & -\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)})\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(3)}]+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]\\ & & -\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]\end{array}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}. On the other hand, by using [p(x¯3),p(y¯3)]=−[p(y¯3),p(x¯3)]\left[p\left({\bar{x}}_{3}),p({\bar{y}}_{3})]=-\left[p({\bar{y}}_{3}),p\left({\bar{x}}_{3})], we obtain from aforementioned identity F[p(x¯3),p(y¯3)]=∑π∈S3∑σ∈S3F([xπ(1)xπ(2),yσ(1)yσ(2)])xπ(3)yσ(3)−∑π∈S3∑σ∈S3[xπ(1),yσ(1)yσ(2)]F(xπ(2))xπ(3)yσ(3)+∑π∈S3∑σ∈S3[xπ(1),yσ(1)yσ(2)]F(xπ(2)xπ(3))yσ(3)−∑π∈S3∑σ∈S3xπ(1)F([xπ(2),yσ(1)yσ(2)])xπ(3)yσ(3)+∑π∈S3∑σ∈S3xπ(1)F([xπ(2)xπ(3),yσ(1)yσ(2)])yσ(3)+∑π∈S3∑σ∈S3F(xπ(1)xπ(2))[xπ(3),yσ(1)yσ(2)]yσ(3)\begin{array}{rcl}F\left[p\left({\bar{x}}_{3}),p({\bar{y}}_{3})]& =& \displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)}){y}_{\sigma \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]{y}_{\sigma \left(3)}\end{array}−∑π∈S3∑σ∈S3xπ(1)F(xπ(2))[xπ(3),yσ(1)yσ(2)]yσ(3)−∑π∈S3∑σ∈S3[xπ(1)xπ(2)xπ(3),yσ(1)]F(yσ(2))yσ(3)+∑π∈S3∑σ∈S3[xπ(1)xπ(2)xπ(3),yσ(1)]F(yσ(2)yσ(3))−∑π∈S3∑σ∈S3yσ(1)F([xπ(1)xπ(2),yσ(2)])xπ(3)yσ(3)+∑π∈S3∑σ∈S3yσ(1)[xπ(1),yσ(2)]F(xπ(2))xπ(3)yσ(3)−∑π∈S3∑σ∈S3yσ(1)[xπ(1),yσ(2)]F(xπ(2)xπ(3))yσ(3)+∑π∈S3∑σ∈S3yσ(1)xπ(1)F([xπ(2),yσ(2)])xπ(3)yσ(3)−∑π∈S3∑σ∈S3yσ(1)xπ(1)F([xπ(2)xπ(3),yσ(2)])yσ(3)−∑π∈S3∑σ∈S3yσ(1)F(xπ(1)xπ(2))[xπ(3),yσ(2)]yσ(3)+∑π∈S3∑σ∈S3yσ(1)xπ(1)F(xπ(2))[xπ(3),yσ(2)]yσ(3)+∑π∈S3∑σ∈S3yσ(1)F([xπ(1)xπ(2),yσ(2)yσ(3)])xπ(3)−∑π∈S3∑σ∈S3yσ(1)[xπ(1),yσ(2)yσ(3)]F(xπ(2))xπ(3)+∑π∈S3∑σ∈S3yσ(1)[xπ(1),yσ(2)yσ(3)]F(xπ(2)xπ(3))−∑π∈S3∑σ∈S3yσ(1)xπ(1)F([xπ(2),yσ(2)yσ(3)])xπ(3)+∑π∈S3∑σ∈S3yσ(1)xπ(1)F([xπ(2)xπ(3),yσ(2)yσ(3)])+∑π∈S3∑σ∈S3yσ(1)F(xπ(1)xπ(2))[xπ(3),yσ(2)yσ(3)]−∑π∈S3∑σ∈S3yσ(1)xπ(1)F(xπ(2))[xπ(3),yσ(2)yσ(3)]+∑π∈S3∑σ∈S3F(yσ(1)yσ(2))[xπ(1)xπ(2)xπ(3),yσ(3)]−∑π∈S3∑σ∈S3yσ(1)F(yσ(2))[xπ(1)xπ(2)xπ(3),yσ(3)]\begin{array}{rcl}& & -\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]{y}_{\sigma \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)})-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}\left[{x}_{\pi \left(1)},{y}_{\sigma \left(2)}]F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}\left[{x}_{\pi \left(1)},{y}_{\sigma \left(2)}]F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)}){y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}\\ & & -\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},{y}_{\sigma \left(2)}]{y}_{\sigma \left(3)}+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},{y}_{\sigma \left(2)}]{y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]){x}_{\pi \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}\left[{x}_{\pi \left(1)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}\left[{x}_{\pi \left(1)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)})-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]){x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}])+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]\\ & & -\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)})\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(3)}]\\ & & -\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)})\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(3)}]\end{array}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}. By comparing so obtained identities, we arrive at (13)0=∑π∈S3∑σ∈S3F([xπ(1)xπ(2),yσ(1)yσ(2)])yσ(3)xπ(3)−F(xπ(1)xπ(2))yσ(1)yσ(2)yσ(3)xπ(3)+F(yσ(1)yσ(2))xπ(1)xπ(2)yσ(3)xπ(3)−yσ(1)F(yσ(2))xπ(1)xπ(2)yσ(3)xπ(3)−xπ(1)F(yσ(1)yσ(2))xπ(2)yσ(3)xπ(3)−yσ(1)xπ(1)F([yσ(2)yσ(3),xπ(2)])xπ(3)+xπ(1)yσ(1)F(yσ(2))xπ(2)yσ(3)xπ(3)+xπ(1)F(xπ(2))yσ(1)yσ(2)yσ(3)xπ(3)+yσ(1)F(xπ(1)xπ(2))yσ(2)yσ(3)xπ(3)−yσ(1)xπ(1)F(xπ(2))yσ(2)yσ(3)xπ(3)+∑π∈S3∑σ∈S3F([yσ(1)yσ(2),xπ(1)xπ(2)])xπ(3)yσ(3)−F(yσ(1)yσ(2))xπ(1)xπ(2)xπ(3)yσ(3)+F(xπ(1)xπ(2))yσ(1)yσ(2)xπ(3)yσ(3)−xπ(1)F(xπ(2))yσ(1)yσ(2)xπ(3)yσ(3)−yσ(1)F(xπ(1)xπ(2))yσ(2)xπ(3)yσ(3)−xπ(1)yσ(1)F([xπ(2)xπ(3),yσ(2)])yσ(3)+yσ(1)xπ(1)F(xπ(2))yσ(2)xπ(3)yσ(3)+yσ(1)F(yσ(2))xπ(1)xπ(2)xπ(3)yσ(3)+xπ(1)F(yσ(1)yσ(2))xπ(2)xπ(3)yσ(3)−xπ(1)yσ(1)F(yσ(2))xπ(2)xπ(3)yσ(3)+∑π∈S3∑σ∈S3xπ(1)yσ(1)F([xπ(2)xπ(3),yσ(2)yσ(3)])+xπ(1)yσ(1)xπ(2)xπ(3)F(yσ(2))yσ(3)−xπ(1)yσ(1)xπ(2)xπ(3)F(yσ(2)yσ(3))+xπ(1)yσ(1)yσ(2)yσ(3)F(xπ(2)xπ(3))−xπ(1)yσ(1)xπ(2)F(yσ(2))yσ(3)xπ(3)+xπ(1)yσ(1)xπ(2)F(yσ(2)yσ(3))xπ(3)−xπ(1)yσ(1)yσ(2)yσ(3)F(xπ(2))xπ(3)−xπ(1)F([xπ(2),yσ(1)yσ(2)])yσ(3)xπ(3)+xπ(1)yσ(1)F([xπ(2),yσ(2)])yσ(3)xπ(3)−xπ(1)yσ(1)F([xπ(2),yσ(2)yσ(3)])xπ(3)−xπ(1)F([yσ(1)yσ(2),xπ(2)])xπ(3)yσ(3)+xπ(1)yσ(1)yσ(2)F(xπ(2))xπ(3)yσ(3)−xπ(1)yσ(1)yσ(2)F(xπ(2)xπ(3))yσ(3)+∑π∈S3∑σ∈S3yσ(1)xπ(1)F([yσ(2)yσ(3),xπ(2)xπ(3)])+yσ(1)xπ(1)yσ(2)yσ(3)F(xπ(2))xπ(3)−yσ(1)xπ(1)yσ(2)yσ(3)F(xπ(2)xπ(3))+yσ(1)xπ(1)xπ(2)xπ(3)F(yσ(2)yσ(3))−yσ(1)xπ(1)yσ(2)F(xπ(2))xπ(3)yσ(3)+yσ(1)xπ(1)yσ(2)F(xπ(2)xπ(3))yσ(3)−yσ(1)xπ(1)xπ(2)xπ(3)F(yσ(2))yσ(3)\hspace{-36.15em}\begin{array}{rcl}0& =& \displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}-F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){y}_{\sigma \left(1)}{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(1)}{x}_{\pi \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}-{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(1)}{x}_{\pi \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & -{x}_{\pi \left(1)}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}-{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(2)}{y}_{\sigma \left(3)},{x}_{\pi \left(2)}]){x}_{\pi \left(3)}\\ & & +{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){y}_{\sigma \left(1)}{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +{y}_{\sigma \left(1)}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){y}_{\sigma \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}-{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){y}_{\sigma \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F\left([{y}_{\sigma \left(1)}{y}_{\sigma \left(2)},{x}_{\pi \left(1)}{x}_{\pi \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}-F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(1)}{x}_{\pi \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & +F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){y}_{\sigma \left(1)}{y}_{\sigma \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){y}_{\sigma \left(1)}{y}_{\sigma \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & -{y}_{\sigma \left(1)}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){y}_{\sigma \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}\\ & & +{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){y}_{\sigma \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(1)}{x}_{\pi \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & +{x}_{\pi \left(1)}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}])\\ & & +{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{x}_{\pi \left(2)}{x}_{\pi \left(3)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}-{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{x}_{\pi \left(2)}{x}_{\pi \left(3)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)})\\ & & +{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)})-{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)}){x}_{\pi \left(3)}-{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}\\ & & -{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & -{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]){x}_{\pi \left(3)}-{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(1)}{y}_{\sigma \left(2)},{x}_{\pi \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & +{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)}){y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(2)}{y}_{\sigma \left(3)},{x}_{\pi \left(2)}{x}_{\pi \left(3)}])\\ & & +{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}-{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)})\\ & & +{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{x}_{\pi \left(2)}{x}_{\pi \left(3)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)})-{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & +{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)}){y}_{\sigma \left(3)}-{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{x}_{\pi \left(2)}{x}_{\pi \left(3)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}\end{array}(13)−yσ(1)F([yσ(2),xπ(1)xπ(2)])xπ(3)yσ(3)+yσ(1)xπ(1)F([yσ(2),xπ(2)])xπ(3)yσ(3)−yσ(1)xπ(1)F([yσ(2),xπ(2)xπ(3)])yσ(3)−yσ(1)F([xπ(1)xπ(2),yσ(2)])yσ(3)xπ(3)−yσ(1)xπ(1)xπ(2)F(yσ(2)yσ(3))xπ(3)+yσ(1)xπ(1)xπ(2)F(yσ(2))yσ(3)xπ(3)\begin{array}{rcl}& & -{y}_{\sigma \left(1)}F\left([{y}_{\sigma \left(2)},{x}_{\pi \left(1)}{x}_{\pi \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(2)},{x}_{\pi \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & -{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(2)},{x}_{\pi \left(2)}{x}_{\pi \left(3)}]){y}_{\sigma \left(3)}-{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & -{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)}){x}_{\pi \left(3)}+{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\end{array}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}. By using the theory of functional identities, we can conclude that (14)∑π∈S2∑σ∈S2F([xπ(1)xπ(2),yσ(1)yσ(2)])−F(xπ(1)xπ(2))yσ(1)yσ(2)+F(yσ(1)yσ(2))xπ(1)xπ(2)−yσ(1)F(yσ(2))xπ(1)xπ(2)+xπ(1)F(xπ(2))yσ(1)yσ(2)=x1p1(x2,y1,y2)+x2p2(x1,y1,y2)+y1p3(x1,x2,y2)+y2p4(x1,x2,y1)+λp(x1,x2,y1,y2)\begin{array}{l}\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{2}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{2}}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}])-F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){y}_{\sigma \left(1)}{y}_{\sigma \left(2)}+F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(1)}{x}_{\pi \left(2)}\\ -{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(1)}{x}_{\pi \left(2)}+{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){y}_{\sigma \left(1)}{y}_{\sigma \left(2)}\\ \hspace{1.0em}={x}_{1}{p}_{1}\left({x}_{2},{y}_{1},{y}_{2})+{x}_{2}{p}_{2}\left({x}_{1},{y}_{1},{y}_{2})+{y}_{1}{p}_{3}\left({x}_{1},{x}_{2},{y}_{2})+{y}_{2}{p}_{4}\left({x}_{1},{x}_{2},{y}_{1})+{\lambda }_{p}\left({x}_{1},{x}_{2},{y}_{1},{y}_{2})\end{array}and (15)∑π∈S2∑σ∈S2F([xπ(2)xπ(3),yσ(2)yσ(3)])−xπ(2)xπ(3)F(yσ(2)yσ(3))+yσ(2)yσ(3)F(xπ(2)xπ(3))+xπ(2)xπ(3)F(yσ(2))yσ(3)−yσ(2)yσ(3)F(xπ(2))xπ(3)=q1(x3,y2,y3)x2+q2(x2,y2,y3)x3+q3(x2,x3,y3)y2+q4(x2,x3,y2)y3+λq(x2,x3,y2,y3).\hspace{-45em}\begin{array}{l}\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{2}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{2}}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}])-{x}_{\pi \left(2)}{x}_{\pi \left(3)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)})+{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)})+{x}_{\pi \left(2)}{x}_{\pi \left(3)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}\hspace{1em}-{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}={q}_{1}\left({x}_{3},{y}_{2},{y}_{3}){x}_{2}+{q}_{2}\left({x}_{2},{y}_{2},{y}_{3}){x}_{3}+{q}_{3}\left({x}_{2},{x}_{3},{y}_{3}){y}_{2}+{q}_{4}\left({x}_{2},{x}_{3},{y}_{2}){y}_{3}+{\lambda }_{q}\left({x}_{2},{x}_{3},{y}_{2},{y}_{3}).\end{array}We also have ∑π∈S2∑σ∈S2F([yσ(1)yσ(2),xπ(1)xπ(2)])−F(yσ(1)yσ(2))xπ(1)xπ(2)+F(xπ(1)xπ(2))yσ(1)yσ(2)−xπ(1)F(xπ(2))yσ(1)yσ(2)+yσ(1)F(yσ(2))xπ(1)xπ(2)=x1p1′(x2,y1,y2)+x2p2′(x1,y1,y2)+y1p3′(x1,x2,y2)+y2p4′(x1,x2,y1)+λp′(x1,x2,y1,y2)\begin{array}{l}\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{2}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{2}}F\left([{y}_{\sigma \left(1)}{y}_{\sigma \left(2)},{x}_{\pi \left(1)}{x}_{\pi \left(2)}])-F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(1)}{x}_{\pi \left(2)}+F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){y}_{\sigma \left(1)}{y}_{\sigma \left(2)}\\ \hspace{1.0em}-{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){y}_{\sigma \left(1)}{y}_{\sigma \left(2)}+{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(1)}{x}_{\pi \left(2)}\\ \hspace{1.0em}={x}_{1}{p}_{1}^{^{\prime} }\left({x}_{2},{y}_{1},{y}_{2})+{x}_{2}{p}_{2}^{^{\prime} }\left({x}_{1},{y}_{1},{y}_{2})+{y}_{1}{p}_{3}^{^{\prime} }\left({x}_{1},{x}_{2},{y}_{2})+{y}_{2}{p}_{4}^{^{\prime} }\left({x}_{1},{x}_{2},{y}_{1})+{\lambda }_{p^{\prime} }\left({x}_{1},{x}_{2},{y}_{1},{y}_{2})\end{array}and ∑π∈S2∑σ∈S2F([yσ(2)yσ(3),xπ(2)xπ(3)])−yσ(2)yσ(3)F(xπ(2)xπ(3))+xπ(2)xπ(3)F(yσ(2)yσ(3))−xπ(2)xπ(3)F(yσ(2))yσ(3)+yσ(2)yσ(3)F(xπ(2))xπ(3)=q1′(x3,y2,y3)x2+q2′(x2,y2,y3)x3+q3′(x2,x3,y3)y2+q4′(x2,x3,y2)y3+λq′(x2,x3,y2,y3)\begin{array}{l}\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{2}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{2}}F\left([{y}_{\sigma \left(2)}{y}_{\sigma \left(3)},{x}_{\pi \left(2)}{x}_{\pi \left(3)}])-{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)})+{x}_{\pi \left(2)}{x}_{\pi \left(3)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)})\\ \hspace{1.0em}-{x}_{\pi \left(2)}{x}_{\pi \left(3)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}+{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}\\ \hspace{1.0em}={q}_{1}^{^{\prime} }\left({x}_{3},{y}_{2},{y}_{3}){x}_{2}+{q}_{2}^{^{\prime} }\left({x}_{2},{y}_{2},{y}_{3}){x}_{3}+{q}_{3}^{^{\prime} }\left({x}_{2},{x}_{3},{y}_{3}){y}_{2}+{q}_{4}^{^{\prime} }\left({x}_{2},{x}_{3},{y}_{2}){y}_{3}+{\lambda }_{q^{\prime} }\left({x}_{2},{x}_{3},{y}_{2},{y}_{3})\end{array}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}and pi,pi′,qi,qi′:ℒ3→R{p}_{i},{p}_{i}^{^{\prime} },{q}_{i},{q}_{i}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{3}\to R, i=1,2,3,4i=1,2,3,4and λp,λp′,λq,λq′:ℒ4→C(ℒ){\lambda }_{p},{\lambda }_{p^{\prime} },{\lambda }_{q},{\lambda }_{q^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{4}\to C\left({\mathcal{ {\mathcal L} }}). By comparing identities (14) and (15), we arrive at (16)∑π∈S2∑σ∈S2(−F(xπ(1)xπ(2))yσ(1)yσ(2)+F(yσ(1)yσ(2))xπ(1)xπ(2)−yσ(1)F(yσ(2))xπ(1)xπ(2)+xπ(1)F(xπ(2))yσ(1)yσ(2)+xπ(1)xπ(2)F(yσ(1)yσ(2))−yσ(1)yσ(2)F(xπ(1)xπ(2))+yσ(1)yσ(2)F(xπ(1))xπ(2)−xπ(1)xπ(2)F(yσ(1))yσ(2))=xπ(1)p1(xπ(2),yσ(1),yσ(2))+xπ(2)p2(xπ(1),yσ(1),yσ(2))+yσ(1)p3(xπ(1),xπ(2),yσ(2))+yσ(2)p4(xπ(1),xπ(2),yσ(1))+λ1(xπ(1),xπ(2),yσ(1),yσ(2))−q1(xπ(2),yσ(1),yσ(2))xπ(1)−q2(xπ(1),yσ(1),yσ(2))xπ(2)−q3(xπ(1),xπ(2),yσ(2))yσ(1)−q4(xπ(1),xπ(2),yσ(1))yσ(2)−μ1(xπ(1),xπ(2),yσ(1),yσ(2))\hspace{-25em}\begin{array}{l}\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{2}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{2}}(-F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){y}_{\sigma \left(1)}{y}_{\sigma \left(2)}+F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(1)}{x}_{\pi \left(2)}-{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(1)}{x}_{\pi \left(2)}+{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){y}_{\sigma \left(1)}{y}_{\sigma \left(2)}\\ \hspace{1.0em}+{x}_{\pi \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)})-{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)})+{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(1)}){x}_{\pi \left(2)}-{x}_{\pi \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(1)}){y}_{\sigma \left(2)})\\ \hspace{1.0em}={x}_{\pi \left(1)}{p}_{1}\left({x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)})+{x}_{\pi \left(2)}{p}_{2}\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)})+{y}_{\sigma \left(1)}{p}_{3}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(2)})\\ \hspace{2.0em}+{y}_{\sigma \left(2)}{p}_{4}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)})+{\lambda }_{1}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)})-{q}_{1}\left({x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){x}_{\pi \left(1)}\\ \hspace{2.0em}-{q}_{2}\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){x}_{\pi \left(2)}-{q}_{3}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(2)}){y}_{\sigma \left(1)}-{q}_{4}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)}){y}_{\sigma \left(2)}-{\mu }_{1}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)})\end{array}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}and pi,pi′,qi,qi′:ℒ3→R{p}_{i},{p}_{i}^{^{\prime} },{q}_{i},{q}_{i}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{3}\to R, i=1,2,3,4i=1,2,3,4and λp,λp′,λq,λq′:ℒ4→C(ℒ){\lambda }_{p},{\lambda }_{p^{\prime} },{\lambda }_{q},{\lambda }_{q^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{4}\to C\left({\mathcal{ {\mathcal L} }}). By using equation (16) and the theory of functional identities, it follows that ∑σ∈S2(F(yσ(1)yσ(2))xπ(1)−yσ(1)F(yσ(2))xπ(1))+q2(xπ(1),yσ(1),yσ(2))=xπ(1)m1(yσ(1),yσ(2))+yσ(1)m2(xπ(1),yσ(2))+yσ(2)m3(xπ(1),yσ(1))+λm(xπ(1),yσ(1),yσ(2))\begin{array}{l}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{2}}\left(F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(1)}-{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(1)})+{q}_{2}\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)})\\ \hspace{1.0em}={x}_{\pi \left(1)}{m}_{1}({y}_{\sigma \left(1)},{y}_{\sigma \left(2)})+{y}_{\sigma \left(1)}{m}_{2}\left({x}_{\pi \left(1)},{y}_{\sigma \left(2)})+{y}_{\sigma \left(2)}{m}_{3}\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)})+{\lambda }_{m}\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)})\end{array}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}, q2:ℒ3→R{q}_{2}:{{\mathcal{ {\mathcal L} }}}^{3}\to R, mi:ℒ2→R{m}_{i}:{{\mathcal{ {\mathcal L} }}}^{2}\to R, i=1,2,3i=1,2,3and λm:ℒ3→C(ℒ){\lambda }_{m}:{{\mathcal{ {\mathcal L} }}}^{3}\to C\left({\mathcal{ {\mathcal L} }}). Now setting xπ(1),yσ(1),yσ(2)=x{x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}=xin the aforementioned equation, we obtain (17)2F(x2)x−2xF(x)x+q2(x,x,x)=xm1(x,x)+xm2(x,x)+xm3(x,x)+λm(x,x,x)2F\left({x}^{2})x-2xF\left(x)x+{q}_{2}\left(x,x,x)=x{m}_{1}\left(x,x)+x{m}_{2}\left(x,x)+x{m}_{3}\left(x,x)+{\lambda }_{m}\left(x,x,x)for all x∈Rx\in R.On the other hand, using equation (16) and the theory of functional identities, we arrive at ∑σ∈S2(xπ(1)F(yσ(1)yσ(2))−xπ(1)F(yσ(1))yσ(2))−p2(xπ(1),yσ(1),yσ(2))=n1′(yσ(1),yσ(2))xπ(1)+n2′(xπ(1),yσ(2))yσ(1)+n3′(xπ(1),yσ(1))yσ(2)+λn′(xπ(1),yσ(1),yσ(2))\begin{array}{l}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{2}}\left({x}_{\pi \left(1)}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)})-{x}_{\pi \left(1)}F({y}_{\sigma \left(1)}){y}_{\sigma \left(2)})-{p}_{2}\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)})\\ \hspace{1.0em}={n}_{1}^{^{\prime} }({y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){x}_{\pi \left(1)}+{n}_{2}^{^{\prime} }\left({x}_{\pi \left(1)},{y}_{\sigma \left(2)}){y}_{\sigma \left(1)}+{n}_{3}^{^{\prime} }\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)}){y}_{\sigma \left(2)}+{\lambda }_{n}^{^{\prime} }\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)})\end{array}x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}, p2:ℒ3→R{p}_{2}:{{\mathcal{ {\mathcal L} }}}^{3}\to R, ni′:ℒ2→R{n}_{i}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{2}\to R, i=1,2,3i=1,2,3and λn′:ℒ3→C(ℒ){\lambda }_{n}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{3}\to C\left({\mathcal{ {\mathcal L} }}).Now setting xπ(1),yσ(1),yσ(2)=x{x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}=x, we obtain (18)2xF(x2)−2xF(x)x−p2(x,x,x)=n1′(x,x)x+n2′(x,x)x+n3′(x,x)x+λn′(x,x,x)2xF\left({x}^{2})-2xF\left(x)x-{p}_{2}\left(x,x,x)={n}_{1}^{^{\prime} }\left(x,x)x+{n}_{2}^{^{\prime} }\left(x,x)x+{n}_{3}^{^{\prime} }\left(x,x)x+{\lambda }_{n}^{^{\prime} }\left(x,x,x)for all x∈Rx\in R. Equation (13) can now be rewritten as follows: 0=∑π∈S3∑σ∈S3(xπ(1)p1(xπ(2),yσ(1),yσ(2))yσ(3)xπ(3)+xπ(2)p2(xπ(1),yσ(1),yσ(2))yσ(3)xπ(3)+yσ(1)p3(xπ(1),xπ(2),yσ(2))yσ(3)xπ(3)+yσ(2)p4(xπ(1),xπ(2),yσ(1))yσ(3)xπ(3)+λp(xπ(1),xπ(2),yσ(1),yσ(2))yσ(3)xπ(3)+xπ(1)yσ(1)F(yσ(2))xπ(2)yσ(3)xπ(3)−xπ(1)F(yσ(1)yσ(2))xπ(2)yσ(3)xπ(3)−yσ(1)xπ(1)F([yσ(2)yσ(3),xπ(2)])xπ(3)+yσ(1)F(xπ(1)xπ(2))yσ(2)yσ(3)xπ(3)−yσ(1)xπ(1)F(xπ(2))yσ(2)yσ(3)xπ(3)+∑π∈S3∑σ∈S3(xπ(1)p1′(xπ(2),yσ(1),yσ(2))xπ(3)yσ(3)+xπ(2)p2′(xπ(1),yσ(1),yσ(2))xπ(3)yσ(3)+yσ(1)p3′(xπ(1),xπ(2),yσ(2))xπ(3)yσ(3)+yσ(2)p4′(xπ(1),xπ(2),yσ(1))xπ(3)yσ(3))+λp′(xπ(1),xπ(2),yσ(1),yσ(2))xπ(3)yσ(3)+yσ(1)xπ(1)F(xπ(2))yσ(2)xπ(3)yσ(3)−yσ(1)F(xπ(1)xπ(2))yσ(2)xπ(3)yσ(3)−xπ(1)yσ(1)F([xπ(2)xπ(3),yσ(2)])yσ(3)+xπ(1)F(yσ(1)yσ(2))xπ(2)xπ(3)yσ(3)−xπ(1)yσ(1)F(yσ(2))xπ(2)xπ(3)yσ(3))+∑π∈S3∑σ∈S3(xπ(1)yσ(1)q1(xπ(3),yσ(2),yσ(3))xπ(2)+xπ(1)yσ(1)q2(xπ(2),yσ(2),yσ(3))xπ(3)+xπ(1)yσ(1)q3(xπ(2),xπ(3),yσ(3))yσ(2)+xπ(1)yσ(1)q4(xπ(2),xπ(3),yσ(2))yσ(3)+xπ(1)yσ(1)λq(xπ(2),xπ(3),yσ(2),yσ(3))−xπ(1)yσ(1)xπ(2)F(yσ(2))yσ(3)xπ(3)+xπ(1)yσ(1)xπ(2)F(yσ(2)yσ(3))xπ(3)−xπ(1)F([xπ(2),yσ(1)yσ(2)])yσ(3)xπ(3)+xπ(1)yσ(1)F([xπ(2),yσ(2)])yσ(3)xπ(3)−xπ(1)yσ(1)F([xπ(2),yσ(2)yσ(3)])xπ(3)−xπ(1)F([yσ(1)yσ(2),xπ(2)])xπ(3)yσ(3)+xπ(1)yσ(1)yσ(2)F(xπ(2))xπ(3)yσ(3)−xπ(1)yσ(1)yσ(2)F(xπ(2)xπ(3))yσ(3))+∑π∈S3∑σ∈S3(yσ(1)xπ(1)q1′(xπ(3),yσ(2),yσ(3))xπ(2)+yσ(1)xπ(1)q2′(xπ(2),yσ(2),yσ(3))xπ(3)+yσ(1)xπ(1)q3′(xπ(2),xπ(3),yσ(3))yσ(2)+yσ(1)xπ(1)q4′(xπ(2),xπ(3),yσ(2))yσ(3)+yσ(1)xπ(1)λq′(xπ(2),xπ(3),yσ(2),yσ(3))−yσ(1)xπ(1)yσ(2)F(xπ(2))xπ(3)yσ(3)+yσ(1)xπ(1)yσ(2)F(xπ(2)xπ(3))yσ(3)−yσ(1)F([yσ(2),xπ(1)xπ(2)])xπ(3)yσ(3)+yσ(1)xπ(1)F([yσ(2),xπ(2)])xπ(3)yσ(3)−yσ(1)xπ(1)F([yσ(2),xπ(2)xπ(3)])yσ(3)−yσ(1)F([xπ(1)xπ(2),yσ(2)])yσ(3)xπ(3)−yσ(1)xπ(1)xπ(2)F(yσ(2)yσ(3))xπ(3)+yσ(1)xπ(1)xπ(2)F(yσ(2))yσ(3)xπ(3))\hspace{-36.75em}\begin{array}{rcl}0& =& \displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}({x}_{\pi \left(1)}{p}_{1}\left({x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{x}_{\pi \left(2)}{p}_{2}\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +{y}_{\sigma \left(1)}{p}_{3}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{y}_{\sigma \left(2)}{p}_{4}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +{\lambda }_{p}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & -{x}_{\pi \left(1)}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}-{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(2)}{y}_{\sigma \left(3)},{x}_{\pi \left(2)}]){x}_{\pi \left(3)}\\ & & +{y}_{\sigma \left(1)}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){y}_{\sigma \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}-{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){y}_{\sigma \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}({x}_{\pi \left(1)}{p}_{1}^{^{\prime} }\left({x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{x}_{\pi \left(2)}{p}_{2}^{^{\prime} }\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & +{y}_{\sigma \left(1)}{p}_{3}^{^{\prime} }\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{y}_{\sigma \left(2)}{p}_{4}^{^{\prime} }\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)})\\ & & +{\lambda }_{p^{\prime} }\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){y}_{\sigma \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & -{y}_{\sigma \left(1)}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){y}_{\sigma \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}\\ & & +{x}_{\pi \left(1)}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)})\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}({x}_{\pi \left(1)}{y}_{\sigma \left(1)}{q}_{1}\left({x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)}){x}_{\pi \left(2)}\\ & & +{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{q}_{2}\left({x}_{\pi \left(2)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)}){x}_{\pi \left(3)}+{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{q}_{3}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(3)}){y}_{\sigma \left(2)}\\ & & +{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{q}_{4}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}+{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{\lambda }_{q}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)})\\ & & -{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)}){x}_{\pi \left(3)}\\ & & -{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & -{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]){x}_{\pi \left(3)}-{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(1)}{y}_{\sigma \left(2)},{x}_{\pi \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & +{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)}){y}_{\sigma \left(3)})\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}({y}_{\sigma \left(1)}{x}_{\pi \left(1)}{q}_{1}^{^{\prime} }\left({x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)}){x}_{\pi \left(2)}\\ & & +{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{q}_{2}^{^{\prime} }\left({x}_{\pi \left(2)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)}){x}_{\pi \left(3)}+{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{q}_{3}^{^{\prime} }\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(3)}){y}_{\sigma \left(2)}\\ & & +{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{q}_{4}^{^{\prime} }\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}+{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{\lambda }_{q^{\prime} }\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)})\\ & & -{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)}){y}_{\sigma \left(3)}\\ & & -{y}_{\sigma \left(1)}F\left([{y}_{\sigma \left(2)},{x}_{\pi \left(1)}{x}_{\pi \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(2)},{x}_{\pi \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & -{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(2)},{x}_{\pi \left(2)}{x}_{\pi \left(3)}]){y}_{\sigma \left(3)}-{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & -{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)}){x}_{\pi \left(3)}+{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)})\end{array}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}, pi,pi′,qi,qi′:ℒ3→R{p}_{i},{p}_{i}^{^{\prime} },{q}_{i},{q}_{i}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{3}\to R, i=1,2,3,4i=1,2,3,4and λp,λp′,λq,λq′:ℒ4→C(ℒ){\lambda }_{p},{\lambda }_{p^{\prime} },{\lambda }_{q},{\lambda }_{q^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{4}\to C\left({\mathcal{ {\mathcal L} }}). By using the theory of functional identities and exposing everything from the same side (left), the aforementioned equation can now be rewritten as follows: (19)0=∑π∈S3∑σ∈S3xπ(1)(p1(xπ(2),yσ(1),yσ(2))yσ(3)xπ(3)−F(yσ(1)yσ(2))xπ(2)yσ(3)xπ(3)+yσ(1)F(yσ(2))xπ(2)yσ(3)xπ(3)+p1′(xπ(2),yσ(1),yσ(2))xπ(3)yσ(3)−yσ(1)F([xπ(2)xπ(3),yσ(2)])yσ(3)+F(yσ(1)yσ(2))xπ(2)xπ(3)yσ(3)−yσ(1)F(yσ(2))xπ(2)xπ(3)yσ(3)+yσ(1)q1(xπ(3),yσ(2),yσ(3))xπ(2)\begin{array}{rcl}0& =& \displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}({p}_{1}\left({x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}-F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{p}_{1}^{^{\prime} }\left({x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}\\ & & +F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{y}_{\sigma \left(1)}{q}_{1}\left({x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)}){x}_{\pi \left(2)}\end{array}(19)+yσ(1)q2(xπ(2),yσ(2),yσ(3))xπ(3)+yσ(1)q3(xπ(2),xπ(3),yσ(3))yσ(2)+yσ(1)q4(xπ(2),xπ(3),yσ(2))yσ(3)+yσ(1)λq(xπ(2),xπ(3),yσ(2),yσ(3))−yσ(1)xπ(2)F(yσ(2))yσ(3)xπ(3)+yσ(1)xπ(2)F(yσ(2)yσ(3))xπ(3)−F([xπ(2),yσ(1)yσ(2)])yσ(3)xπ(3)+yσ(1)F([xπ(2),yσ(2)])yσ(3)xπ(3)−yσ(1)F([xπ(2),yσ(2)yσ(3)])xπ(3)−F([yσ(1)yσ(2),xπ(2)])xπ(3)yσ(3)+yσ(1)yσ(2)F(xπ(2))xπ(3)yσ(3)−yσ(1)yσ(2)F(xπ(2)xπ(3))yσ(3))+∑π∈S3∑σ∈S3xπ(2)(p2(xπ(1),yσ(1),yσ(2))yσ(3)xπ(3)+p2′(xπ(1),yσ(1),yσ(2))xπ(3)yσ(3))+∑π∈S3∑σ∈S3xπ(3)(yσ(3)λp′(xπ(1),xπ(2),yσ(1),yσ(2)))+∑π∈S3∑σ∈S3yσ(1)(p3(xπ(1),xπ(2),yσ(2))yσ(3)xπ(3)−xπ(1)F([yσ(2)yσ(3),xπ(2)])xπ(3)+F(xπ(1)xπ(2))yσ(2)yσ(3)xπ(3)−xπ(1)F(xπ(2))yσ(2)yσ(3)xπ(3)+p3′(xπ(1),xπ(2),yσ(2))xπ(3)yσ(3)−F(xπ(1)xπ(2))yσ(2)xπ(3)yσ(3)+xπ(1)F(xπ(2))yσ(2)xπ(3)yσ(3)+xπ(1)q1′(xπ(3),yσ(2),yσ(3))xπ(2)+xπ(1)q2′(xπ(2),yσ(2),yσ(3))xπ(3)+xπ(1)q3′(xπ(2),xπ(3),yσ(3))yσ(2)+xπ(1)q4′(xπ(2),xπ(3),yσ(2))yσ(3)+xπ(1)λq′(xπ(2),xπ(3),yσ(2),yσ(3))−xπ(1)yσ(2)F(xπ(2))xπ(3)yσ(3)+xπ(1)yσ(2)F(xπ(2)xπ(3))yσ(3)−F([yσ(2),xπ(1)xπ(2)])xπ(3)yσ(3)+xπ(1)F([yσ(2),xπ(2)])xπ(3)yσ(3)−xπ(1)F([yσ(2),xπ(2)xπ(3)])yσ(3)−F([xπ(1)xπ(2),yσ(2)])yσ(3)xπ(3)−xπ(1)xπ(2)F(yσ(2)yσ(3))xπ(3)+xπ(1)xπ(2)F(yσ(2))yσ(3)xπ(3))+∑π∈S3∑σ∈S3yσ(2)(p4(xπ(1),xπ(2),yσ(1))yσ(3)xπ(3)+p4′(xπ(1),xπ(2),yσ(1))xπ(3)yσ(3))+∑π∈S3∑σ∈S3yσ(3)(xπ(3)λp(xπ(1),xπ(2),yσ(1),yσ(2)))\begin{array}{rcl}& & +{y}_{\sigma \left(1)}{q}_{2}\left({x}_{\pi \left(2)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)}){x}_{\pi \left(3)}+{y}_{\sigma \left(1)}{q}_{3}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(3)}){y}_{\sigma \left(2)}+{y}_{\sigma \left(1)}{q}_{4}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}\\ & & +{y}_{\sigma \left(1)}{\lambda }_{q}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)})-{y}_{\sigma \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{y}_{\sigma \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)}){x}_{\pi \left(3)}\\ & & -F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}-{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]){x}_{\pi \left(3)}\\ & & -F\left([{y}_{\sigma \left(1)}{y}_{\sigma \left(2)},{x}_{\pi \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)}){y}_{\sigma \left(3)})\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(2)}({p}_{2}\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{p}_{2}^{^{\prime} }\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)})\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(3)}({y}_{\sigma \left(3)}{\lambda }_{p^{\prime} }\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}))\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}({p}_{3}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}-{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(2)}{y}_{\sigma \left(3)},{x}_{\pi \left(2)}]){x}_{\pi \left(3)}\\ & & +F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){y}_{\sigma \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}-{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){y}_{\sigma \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{p}_{3}^{^{\prime} }\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & -F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){y}_{\sigma \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){y}_{\sigma \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{x}_{\pi \left(1)}{q}_{1}^{^{\prime} }\left({x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)}){x}_{\pi \left(2)}\\ & & +{x}_{\pi \left(1)}{q}_{2}^{^{\prime} }\left({x}_{\pi \left(2)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)}){x}_{\pi \left(3)}+{x}_{\pi \left(1)}{q}_{3}^{^{\prime} }\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(3)}){y}_{\sigma \left(2)}+{x}_{\pi \left(1)}{q}_{4}^{^{\prime} }\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}\\ & & +{x}_{\pi \left(1)}{\lambda }_{q^{\prime} }\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)})-{x}_{\pi \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{x}_{\pi \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)}){y}_{\sigma \left(3)}\\ & & -F\left([{y}_{\sigma \left(2)},{x}_{\pi \left(1)}{x}_{\pi \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(2)},{x}_{\pi \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(2)},{x}_{\pi \left(2)}{x}_{\pi \left(3)}]){y}_{\sigma \left(3)}\\ & & -F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}-{x}_{\pi \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)}){x}_{\pi \left(3)}+{x}_{\pi \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)})\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(2)}({p}_{4}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{p}_{4}^{^{\prime} }\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)})\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(3)}\left({x}_{\pi \left(3)}{\lambda }_{p}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}))\end{array}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}, pi,pi′,qi,qi′:ℒ3→R{p}_{i},{p}_{i}^{^{\prime} },{q}_{i},{q}_{i}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{3}\to R, i=1,2,3,4i=1,2,3,4and λp,λp′,λq,λq′:ℒ4→C(ℒ){\lambda }_{p},{\lambda }_{p^{\prime} },{\lambda }_{q},{\lambda }_{q^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{4}\to C\left({\mathcal{ {\mathcal L} }}). Now by using the theory of functional identities and exposing x2{x}_{2}from the left side, we obtain 0=∑π∈S2∑σ∈S3p2(xπ(1),yσ(1),yσ(2))yσ(3)xπ(3)+p2′(xπ(1),yσ(1),yσ(2))xπ(3)yσ(3)0=\sum _{\pi \in {{\mathbb{S}}}_{2}}\sum _{\sigma \in {{\mathbb{S}}}_{3}}{p}_{2}\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{p}_{2}^{^{\prime} }\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}and p2,p2′:ℒ3→R{p}_{2},{p}_{2}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{3}\to R. Again by using the theory of functional identities and exposing everything from the right side, we obtain 0=∑σ∈S2p2(xπ(1),yσ(1),yσ(2))\hspace{-12.55em}0=\sum _{\sigma \in {{\mathbb{S}}}_{2}}{p}_{2}\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)})and 0=∑σ∈S2p2′(xπ(1),yσ(1),yσ(2)).\hspace{-12.55em}0=\sum _{\sigma \in {{\mathbb{S}}}_{2}}{p}_{2}^{^{\prime} }\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}).Therefore, 0=2p2(x,x,x)=2p2′(x,x,x)0=2{p}_{2}\left(x,x,x)=2{p}_{2}^{^{\prime} }\left(x,x,x)for all x∈ℒx\in {\mathcal{ {\mathcal L} }}and p2,p2′:ℒ3→R{p}_{2},{p}_{2}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{3}\to R. Equation (18) can now be rewritten as follows: (20)2xF(x2)−2xF(x)x=n1′(x,x)x+n2′(x,x)x+n3′(x,x)x+λn′(x,x,x)2xF\left({x}^{2})-2xF\left(x)x={n}_{1}^{^{\prime} }\left(x,x)x+{n}_{2}^{^{\prime} }\left(x,x)x+{n}_{3}^{^{\prime} }\left(x,x)x+{\lambda }_{n}^{^{\prime} }\left(x,x,x)for all x∈ℒx\in {\mathcal{ {\mathcal L} }}, ni′:ℒ2→R{n}_{i}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{2}\to R, i=1,2,3i=1,2,3and λn′:ℒ3→C(ℒ){\lambda }_{n}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{3}\to C\left({\mathcal{ {\mathcal L} }}). Now using the theory of functional identities and exposing x1{x}_{1}in (19) from the left side, 0=∑π∈S2∑σ∈S3(p1(xπ(2),yσ(1),yσ(2))yσ(3)xπ(3)−F(yσ(1)yσ(2))xπ(2)yσ(3)xπ(3)+yσ(1)F(yσ(2))xπ(2)yσ(3)xπ(3)+p1′(xπ(2),yσ(1),yσ(2))xπ(3)yσ(3)−yσ(1)F([xπ(2)xπ(3),yσ(2)])yσ(3)+F(yσ(1)yσ(2))xπ(2)xπ(3)yσ(3)−yσ(1)F(yσ(2))xπ(2)xπ(3)yσ(3)+yσ(1)q1(xπ(3),yσ(2),yσ(3))xπ(2)+yσ(1)q2(xπ(2),yσ(2),yσ(3))xπ(3)+yσ(1)q3(xπ(2),xπ(3),yσ(3))yσ(2)+yσ(1)q4(xπ(2),xπ(3),yσ(2))yσ(3)+yσ(1)λq(xπ(2),xπ(3),yσ(2),yσ(3))+yσ(1)xπ(2)F(yσ(2)yσ(3))xπ(3)−yσ(1)xπ(2)F(yσ(2))yσ(3)xπ(3)−F([xπ(2),yσ(1)yσ(2)])yσ(3)xπ(3)\begin{array}{rcl}0& =& \displaystyle \sum _{\pi \in {{\mathbb{S}}}_{2}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}({p}_{1}\left({x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}-F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +{p}_{1}^{^{\prime} }\left({x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}+F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & -{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{y}_{\sigma \left(1)}{q}_{1}\left({x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)}){x}_{\pi \left(2)}+{y}_{\sigma \left(1)}{q}_{2}\left({x}_{\pi \left(2)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)}){x}_{\pi \left(3)}\\ & & +{y}_{\sigma \left(1)}{q}_{3}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(3)}){y}_{\sigma \left(2)}+{y}_{\sigma \left(1)}{q}_{4}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}+{y}_{\sigma \left(1)}{\lambda }_{q}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)})\\ & & +{y}_{\sigma \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)}){x}_{\pi \left(3)}-{y}_{\sigma \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}-F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\end{array}+yσ(1)F([xπ(2),yσ(2)])yσ(3)xπ(3)−yσ(1)F([xπ(2),yσ(2)yσ(3)])xπ(3)−F([yσ(1)yσ(2),xπ(2)])xπ(3)yσ(3)+yσ(1)yσ(2)F(xπ(2))xπ(3)yσ(3)−yσ(1)yσ(2)F(xπ(2)xπ(3))yσ(3))\begin{array}{rcl}& & +{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}-{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]){x}_{\pi \left(3)}-F\left([{y}_{\sigma \left(1)}{y}_{\sigma \left(2)},{x}_{\pi \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & +{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)}){y}_{\sigma \left(3)})\end{array}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}and pi,pi′,qi,qi′:ℒ3→R{p}_{i},{p}_{i}^{^{\prime} },{q}_{i},{q}_{i}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{3}\to R, i=1,2,3,4i=1,2,3,4and λp,λp′,λq,λq′:ℒ4→C(ℒ){\lambda }_{p},{\lambda }_{p^{\prime} },{\lambda }_{q},{\lambda }_{q^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{4}\to C\left({\mathcal{ {\mathcal L} }}). The aforementioned equation can now be rewritten as (everything exposing from the right side) follows: 0=∑π∈S2∑σ∈S3(yσ(1)q1(xπ(3),yσ(2),yσ(3)))xπ(2)+∑π∈S2∑σ∈S3(yσ(1)q2(xπ(2),yσ(2),yσ(3))+p1(xπ(2),yσ(1),yσ(2))yσ(3)−F(yσ(1)yσ(2))xπ(2)yσ(3)+yσ(1)F(yσ(2))xπ(2)yσ(3)−F([xπ(2),yσ(1)yσ(2)])yσ(3)−yσ(1)xπ(2)F(yσ(2))yσ(3)+yσ(1)xπ(2)F(yσ(2)yσ(3))+yσ(1)F([xπ(2),yσ(2)])yσ(3)−yσ(1)F([xπ(2),yσ(2)yσ(3)]))xπ(3)+∑π∈S2∑σ∈S3(λq(xπ(2),xπ(3),yσ(2),yσ(3)))yσ(1)+∑π∈S2∑σ∈S3(yσ(1)q3(xπ(2),xπ(3),yσ(3)))yσ(2)+∑π∈S2∑σ∈S3(p1′(xπ(2),yσ(1),yσ(2))xπ(3)−F([yσ(1)yσ(2),xπ(2)])xπ(3)−yσ(1)F([xπ(2)xπ(3),yσ(2)])+F(yσ(1)yσ(2))xπ(2)xπ(3)−yσ(1)F(yσ(2))xπ(2)xπ(3)+yσ(1)q4(xπ(2),xπ(3),yσ(2))+yσ(1)yσ(2)F(xπ(2))xπ(3)−yσ(1)yσ(2)F(xπ(2)xπ(3)))yσ(3)\begin{array}{rcl}0& =& \displaystyle \sum _{\pi \in {{\mathbb{S}}}_{2}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}({y}_{\sigma \left(1)}{q}_{1}\left({x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)})){x}_{\pi \left(2)}+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{2}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}({y}_{\sigma \left(1)}{q}_{2}\left({x}_{\pi \left(2)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)})+{p}_{1}\left({x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}\\ & & -F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(2)}{y}_{\sigma \left(3)}+{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(2)}{y}_{\sigma \left(3)}-F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}-{y}_{\sigma \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}\\ & & +{y}_{\sigma \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)})+{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}-{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}])){x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{2}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left({\lambda }_{q}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)})){y}_{\sigma \left(1)}+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{2}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left({y}_{\sigma \left(1)}{q}_{3}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(3)})){y}_{\sigma \left(2)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{2}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}({p}_{1}^{^{\prime} }\left({x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){x}_{\pi \left(3)}-F\left([{y}_{\sigma \left(1)}{y}_{\sigma \left(2)},{x}_{\pi \left(2)}]){x}_{\pi \left(3)}-{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(2)}])\\ & & +F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(2)}{x}_{\pi \left(3)}-{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(2)}{x}_{\pi \left(3)}+{y}_{\sigma \left(1)}{q}_{4}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)})+{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}-{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)})){y}_{\sigma \left(3)}\end{array}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}and pi,pi′,qi,qi′:ℒ3→R{p}_{i},{p}_{i}^{^{\prime} },{q}_{i},{q}_{i}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{3}\to R, i=1,2,3,4i=1,2,3,4and λp,λp′,λq,λq′:ℒ4→C(ℒ){\lambda }_{p},{\lambda }_{p^{\prime} },{\lambda }_{q},{\lambda }_{q^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{4}\to C\left({\mathcal{ {\mathcal L} }}). Now by using the theory of functional identities, we obtain from the aforementioned equation: q1(x,x,x)=q3(x,x,x)=0{q}_{1}\left(x,x,x)={q}_{3}\left(x,x,x)=0for all x∈ℒx\in {\mathcal{ {\mathcal L} }}and q1,q3:ℒ3→R{q}_{1},{q}_{3}:{{\mathcal{ {\mathcal L} }}}^{3}\to R. Equation (17) can now be rewritten as follows: (21)2F(x2)x−2xF(x)x=xm1(x,x)+xm2(x,x)+xm3(x,x)+λm(x,x,x)2F\left({x}^{2})x-2xF\left(x)x=x{m}_{1}\left(x,x)+x{m}_{2}\left(x,x)+x{m}_{3}\left(x,x)+{\lambda }_{m}\left(x,x,x)for all x∈ℒx\in {\mathcal{ {\mathcal L} }}, mi:ℒ2→R{m}_{i}:{{\mathcal{ {\mathcal L} }}}^{2}\to R, i=1,2,3i=1,2,3and λm:ℒ3→C(ℒ){\lambda }_{m}:{{\mathcal{ {\mathcal L} }}}^{3}\to C\left({\mathcal{ {\mathcal L} }}). Since ℒ{\mathcal{ {\mathcal L} }}is 6-free, after a finite number of steps using equations (20) and (21), we arrive at 2∑(F(xy)−xF(y))=xf(y)+yg(x)+λ(x,y)2\sum \left(F\left(xy)-xF(y))=xf(y)+yg\left(x)+\lambda \left(x,y)2∑(F(x)y−F(xy))=h(y)x+k(x)y+μ(x,y)2\sum \left(F\left(x)y-F\left(xy))=h(y)x+k\left(x)y+\mu \left(x,y)for all x,y∈Rx,y\in R, f,g,h,k:ℒ→Rf,g,h,k:{\mathcal{ {\mathcal L} }}\to Rand λ,μ:ℒ2→C(ℒ)\lambda ,\mu :{{\mathcal{ {\mathcal L} }}}^{2}\to C\left({\mathcal{ {\mathcal L} }}). Therefore, we obtain (22)2F(xy)+2F(yx)−2xF(y)−2yF(x)=xf(y)+yg(x)+λ(x,y)2F\left(xy)+2F(yx)-2xF(y)-2yF\left(x)=xf(y)+yg\left(x)+\lambda \left(x,y)and (23)2F(x)y+2F(y)x−2F(xy)−2F(yx)=h(y)x+k(x)y+μ(x,y).2F\left(x)y+2F(y)x-2F\left(xy)-2F(yx)=h(y)x+k\left(x)y+\mu \left(x,y).For all, x,y∈Rx,y\in R, f,g,h,k:ℒ→Rf,g,h,k:{\mathcal{ {\mathcal L} }}\to Rand λ,μ:ℒ2→C(ℒ)\lambda ,\mu :{{\mathcal{ {\mathcal L} }}}^{2}\to C\left({\mathcal{ {\mathcal L} }}). Replacing the roles of denotations xxand yyin (22) and comparing so obtained identities leads to 0=xf(y)+yg(x)−yf(x)−xg(y)+λ(x,y)−λ(y,x)0=xf(y)+yg\left(x)-yf\left(x)-xg(y)+\lambda \left(x,y)-\lambda (y,x), which yields f(x)=g(x)f\left(x)=g\left(x)and λ(x,y)=λ(y,x)\lambda \left(x,y)=\lambda (y,x)for all x,y∈ℒx,y\in {\mathcal{ {\mathcal L} }}, f,g:ℒ→Rf,g:{\mathcal{ {\mathcal L} }}\to Rand λ:ℒ2→C(ℒ)\lambda :{{\mathcal{ {\mathcal L} }}}^{2}\to C\left({\mathcal{ {\mathcal L} }}). Putting xxfor yyin (22) leads to (24)4F(x2)=4xF(x)+2xf(x)+λ(x,x).4F\left({x}^{2})=4xF\left(x)+2xf\left(x)+\lambda \left(x,x).Using the same arguments, it follows from (23) that h(x)=k(x)h\left(x)=k\left(x)and μ(x,y)=μ(y,x)\mu \left(x,y)=\mu (y,x)for all x,y∈ℒx,y\in {\mathcal{ {\mathcal L} }}, h,k:ℒ→Rh,k:{\mathcal{ {\mathcal L} }}\to Rand μ:ℒ2→C(ℒ)\mu :{{\mathcal{ {\mathcal L} }}}^{2}\to C\left({\mathcal{ {\mathcal L} }}). Therefore, 4F(x2)=4F(x)x−2k(x)x−μ(x,x).4F\left({x}^{2})=4F\left(x)x-2k\left(x)x-\mu \left(x,x).Comparing the aforementioned relations gives 0=x(4F(x)+2f(x))+(−4(x)+2k(x))x+λ(x,x)+μ(x,x).0=x\left(4F\left(x)+2f\left(x))+\left(-4\left(x)+2k\left(x))x+\lambda \left(x,x)+\mu \left(x,x).Hence, there exists r∈Rr\in Rand λ:ℒ→C(ℒ)\lambda :{\mathcal{ {\mathcal L} }}\to C\left({\mathcal{ {\mathcal L} }})such that 4F(x)+2f(x)=rx+λ(x).\hspace{-15.25em}4F\left(x)+2f\left(x)=rx+\lambda \left(x).Considering 2f(x)=−4F(x)+rx+λ(x)2f\left(x)=-4F\left(x)+rx+\lambda \left(x)in (24) gives (25)4F(x2)=xrx+xλ(x)+λ(x,x).\hspace{-17.65em}4F\left({x}^{2})=xrx+x\lambda \left(x)+\lambda \left(x,x).Replacing yyfor xxand xxfor x2{x}^{2}in (22) gives 4F(x3)=2x2F(x)+2xF(x2)+x2f(x)+xf(x2)+λ(x2,x).4F\left({x}^{3})=2{x}^{2}F\left(x)+2xF\left({x}^{2})+{x}^{2}f\left(x)+xf\left({x}^{2})+\lambda \left({x}^{2},x).Using (7) in the aforementioned relation leads to 4F(x2)x−4xF(x)x+4xF(x2)=2x2F(x)+2xF(x2)+x2f(x)+xf(x2)+λ(x2,x).4F\left({x}^{2})x-4xF\left(x)x+4xF\left({x}^{2})=2{x}^{2}F\left(x)+2xF\left({x}^{2})+{x}^{2}f\left(x)+xf\left({x}^{2})+\lambda \left({x}^{2},x).Using (24) in the aforementioned relation leads to 4xf(x)x+3xλ(x,x)=2xf(x2)+2λ(x2,x).4xf\left(x)x+3x\lambda \left(x,x)=2xf\left({x}^{2})+2\lambda \left({x}^{2},x).Considering 2f(x)=−4F(x)+rx+λ(x)2f\left(x)=-4F\left(x)+rx+\lambda \left(x)and using (25) in the aforementioned relation gives 0=−8xF(x)x+xrx2+x2rx+3x2λ(x)+4xλ(x,x)−xλ(x2)−2λ(x2,x).0=-8xF\left(x)x+xr{x}^{2}+{x}^{2}rx+3{x}^{2}\lambda \left(x)+4x\lambda \left(x,x)-x\lambda \left({x}^{2})-2\lambda \left({x}^{2},x).The complete linearization of this relation and using the theory of functional identities leads to 0=−8F(x)x+rx2+xrx+3xλ(x)+4λ(x,x)−λ(x2)0=-8F\left(x)x+r{x}^{2}+xrx+3x\lambda \left(x)+4\lambda \left(x,x)-\lambda \left({x}^{2})and 0=−8F(x)+rx+xr+3λ(x).\hspace{-14em}0=-8F\left(x)+rx+xr+3\lambda \left(x).Therefore, (26)8F(x)=rx+xr+3λ(x).8F\left(x)=rx+xr+3\lambda \left(x).By substituting the aforementioned equation in (7), we obtain 0=−3λ(x3)−6xλ(x2)−3x2λ(x).0=-3\lambda \left({x}^{3})-6x\lambda \left({x}^{2})-3{x}^{2}\lambda \left(x).Since ℒ{\mathcal{ {\mathcal L} }}is a 6-free subset of RR, the aforementioned identity implies λ(x3)=0\lambda \left({x}^{3})=0, λ(x2)=0\lambda \left({x}^{2})=0, λ(x)=0\lambda \left(x)=0for all x∈Rx\in R. From equation (26), we obtain (27)8F(x2)=rx2+x2r.8F\left({x}^{2})=r{x}^{2}+{x}^{2}r.Right (left) multiplication of the relation (26) by xxgives, respectively, (28)8F(x)x=rx2+xrx8F\left(x)x=r{x}^{2}+xrxand (29)8xF(x)=xrx+x2r.8xF\left(x)=xrx+{x}^{2}r.The relations (27)–(29) imply that the additive mapping FFsatisfies the relation 8F(x2)=8F(x)x+8xF(x)−2xrx.\hspace{-14.45em}8F\left({x}^{2})=8F\left(x)x+8xF\left(x)-2xrx.The aforementioned equation can now be rewritten as follows: 4F(x2)=4F(x)x+4xF(x)−xrx\hspace{-14.45em}4F\left({x}^{2})=4F\left(x)x+4xF\left(x)-xrxand (30)4F(x2)=4F(x)x+4xF(x)−2xqx,4F\left({x}^{2})=4F\left(x)x+4xF\left(x)-2xqx,where r=2qr=2q. Let us now introduce the mapping D:R→RD:R\to Rby (31)D(x)=4F(x)−qx−xq.D\left(x)=4F\left(x)-qx-xq.Obviously, the mapping DDis additive. It is our aim to prove that DDis a Jordan derivation. Putting x2{x}^{2}for xxin the aforementioned relation, we obtain D(x2)=4F(x2)−qx2−x2q,D\left({x}^{2})=4F\left({x}^{2})-q{x}^{2}-{x}^{2}q,which gives, after considering the relation (30), the relation (32)D(x2)=4F(x)x+4xF(x)−2xqx−qx2−x2q.D\left({x}^{2})=4F\left(x)x+4xF\left(x)-2xqx-q{x}^{2}-{x}^{2}q.Right (left) multiplication of the relation (31) by xxgives, respectively, (33)D(x)x=4F(x)x−qx2−xqxD\left(x)x=4F\left(x)x-q{x}^{2}-xqxand (34)xD(x)=4xF(x)−xqx−x2q.xD\left(x)=4xF\left(x)-xqx-{x}^{2}q.The relations (32)–(34) imply that the additive mapping DDsatisfies the relation D(x2)=D(x)x+xD(x)\hspace{-12.15em}D\left({x}^{2})=D\left(x)x+xD\left(x)for all x∈Rx\in R. In other words, DDis a Jordan derivation on RR. According to Herstein theorem, one can conclude that DDis a derivation, which completes the proof of the theorem.□We are now in the position to prove Theorem 4.Proof of Theorem 4The complete linearization of (7) gives us (9). First, suppose that RRis not a PI ring (satisfying the standard polynomial identity of degree less than 6). According to Theorem 5, then there exists q∈Rq\in Rsuch that 4F(x)=D(x)+xq+qx4F\left(x)=D\left(x)+xq+qxfor all x∈ℒx\in {\mathcal{ {\mathcal L} }}.Assume now that RRis a PI ring. It is well known that in this case RRhas a nonzero center (see [20]). Let ccbe a nonzero central element. Picking any x∈Rx\in Rand setting x1=x2=cx{x}_{1}={x}_{2}=cxand x3=x{x}_{3}=xin (9), we obtain 3F(c2x3)=F(c2x2)x+2cF(cx2)x−c2xF(x)x−2cxF(cx)x+xF(c2x2)+2cxF(cx2).3F\left({c}^{2}{x}^{3})=F\left({c}^{2}{x}^{2})x+2cF\left(c{x}^{2})x-{c}^{2}xF\left(x)x-2cxF\left(cx)x+xF\left({c}^{2}{x}^{2})+2cxF\left(c{x}^{2}).Next, setting x1=x2=c{x}_{1}={x}_{2}=cand x3=x3{x}_{3}={x}^{3}in (9), we obtain 3F(c2x3)=F(c2)x3+4cF(cx3)−cx3F(c)−cF(c)x3−c2F(x2)x+c2xF(x)x−c2xF(x2)+x3F(c2)3F\left({c}^{2}{x}^{3})=F\left({c}^{2}){x}^{3}+4cF\left(c{x}^{3})-c{x}^{3}F\left(c)-cF\left(c){x}^{3}-{c}^{2}F\left({x}^{2})x+{c}^{2}xF\left(x)x-{c}^{2}xF\left({x}^{2})+{x}^{3}F\left({c}^{2})for all x∈Rx\in R. Comparing both identities, we obtain 0=F(c2)x3+4cF(cx3)−cx3F(c)−cF(c)x3−c2F(x2)x+c2xF(x)x−c2xF(x2)+x3F(c2).0=F\left({c}^{2}){x}^{3}+4cF\left(c{x}^{3})-c{x}^{3}F\left(c)-cF\left(c){x}^{3}-{c}^{2}F\left({x}^{2})x+{c}^{2}xF\left(x)x-{c}^{2}xF\left({x}^{2})+{x}^{3}F\left({c}^{2}).Next, setting x1=x2=x{x}_{1}={x}_{2}=xand x3=c2{x}_{3}={c}^{2}in (9), we obtain (35)3F(c2x2)=2c2F(x2)+2F(c2x)x−c2F(x)x−xF(c2)x−c2xF(x)+2xF(c2x).3F\left({c}^{2}{x}^{2})=2{c}^{2}F\left({x}^{2})+2F\left({c}^{2}x)x-{c}^{2}F\left(x)x-xF\left({c}^{2})x-{c}^{2}xF\left(x)+2xF\left({c}^{2}x).Next, setting x1=x2=x{x}_{1}={x}_{2}=xand x3=c{x}_{3}=cin (9), we obtain (36)3F(cx2)=2cF(x2)+2F(cx)x−cF(x)x−xF(c)x−cxF(x)+2xF(cx).\hspace{-31.5em}3F\left(c{x}^{2})=2cF\left({x}^{2})+2F\left(cx)x-cF\left(x)x-xF\left(c)x-cxF\left(x)+2xF\left(cx).In case x=cx=c, we arrive at F(c3)=2cF(c2)−c2F(c)F\left({c}^{3})=2cF\left({c}^{2})-{c}^{2}F\left(c). Next, setting x1=x2=c{x}_{1}={x}_{2}=cand x3=x{x}_{3}=xin (9), we obtain (37)3F(c2x)=F(c2)x+4cF(cx)−cxF(c)−c2F(x)−cF(c)x+xF(c2).\hspace{-31.5em}3F\left({c}^{2}x)=F\left({c}^{2})x+4cF\left(cx)-cxF\left(c)-{c}^{2}F\left(x)-cF\left(c)x+xF\left({c}^{2}).Setting x1=x2=c{x}_{1}={x}_{2}=cand x3=x2{x}_{3}={x}^{2}in (9), we obtain 3F(c2x2)=F(c2)x2+4cF(cx2)−cx2F(c)−c2F(x2)−cF(c)x2+x2F(c2).\hspace{-31.5em}3F\left({c}^{2}{x}^{2})=F\left({c}^{2}){x}^{2}+4cF\left(c{x}^{2})-c{x}^{2}F\left(c)-{c}^{2}F\left({x}^{2})-cF\left(c){x}^{2}+{x}^{2}F\left({c}^{2}).From the aforementioned equation and (36), we obtain 9F(c2x2)=3F(c2)x2+8c2F(x2)+8cF(cx)x−4c2F(x)x−4cxF(c)x−4c2xF(x)+8cxF(cx)−3cx2F(c)−3c2F(x2)−3cF(c)x2+3x2F(c2).\begin{array}{rcl}9F\left({c}^{2}{x}^{2})& =& 3F\left({c}^{2}){x}^{2}+8{c}^{2}F\left({x}^{2})+8cF\left(cx)x-4{c}^{2}F\left(x)x-4cxF\left(c)x-4{c}^{2}xF\left(x)+8cxF\left(cx)\\ & & -3c{x}^{2}F\left(c)-3{c}^{2}F\left({x}^{2})-3cF\left(c){x}^{2}+3{x}^{2}F\left({c}^{2}).\end{array}Comparing aforementioned equation with (35) and using (37), we obtain (38)0=c2F(x2)−c2F(x)x−c2xF(x)−F(c2)x2−x2F(c2)+cF(c)x2+xF(c2)x+cx2F(c).0={c}^{2}F\left({x}^{2})-{c}^{2}F\left(x)x-{c}^{2}xF\left(x)-F\left({c}^{2}){x}^{2}-{x}^{2}F\left({c}^{2})+cF\left(c){x}^{2}+xF\left({c}^{2})x+c{x}^{2}F\left(c).Complete linearization of the the aforementioned equation and setting x1=c{x}_{1}=cand x2=x{x}_{2}=x, we obtain 0=2c2F(cx)+c2F(c)x−2c3F(x)+c2xF(c)−cxF(c2)−cF(c2)x.0=2{c}^{2}F\left(cx)+{c}^{2}F\left(c)x-2{c}^{3}F\left(x)+{c}^{2}xF\left(c)-cxF\left({c}^{2})-cF\left({c}^{2})x.Setting c=c2c={c}^{2}in (38) we obtain 0=c4F(x2)−c4F(x)x−c4xF(x)−F(c4)x2−x2F(c4)+c2F(c2)x2+xF(c4)x+c2x2F(c2).\hspace{-39.1em}0={c}^{4}F\left({x}^{2})-{c}^{4}F\left(x)x-{c}^{4}xF\left(x)-F\left({c}^{4}){x}^{2}-{x}^{2}F\left({c}^{4})+{c}^{2}F\left({c}^{2}){x}^{2}+xF\left({c}^{4})x+{c}^{2}{x}^{2}F\left({c}^{2}).On other hand, multiplying equation (38) with c2{c}^{2}, we obtain 0=c4F(x2)−c4F(x)x−c4xF(x)−c2F(c2)x2−c2x2F(c2)+c3F(c)x2+c2xF(c2)x+c3x2F(c).0={c}^{4}F\left({x}^{2})-{c}^{4}F\left(x)x-{c}^{4}xF\left(x)-{c}^{2}F\left({c}^{2}){x}^{2}-{c}^{2}{x}^{2}F\left({c}^{2})+{c}^{3}F\left(c){x}^{2}+{c}^{2}xF\left({c}^{2})x+{c}^{3}{x}^{2}F\left(c).Comparing the aforementioned two equations and using F(c4)=3c2F(c2)−2c3F(c)F\left({c}^{4})=3{c}^{2}F\left({c}^{2})-2{c}^{3}F\left(c), we obtain (39)0=−F(c2)x2−x2F(c2)+cF(c)x2+cx2F(c)+2xF(c2)x−2cxF(c)x.0=-F\left({c}^{2}){x}^{2}-{x}^{2}F\left({c}^{2})+cF\left(c){x}^{2}+c{x}^{2}F\left(c)+2xF\left({c}^{2})x-2cxF\left(c)x.On the other hand, adding the same two equations together, we obtain 0=2c2F(x2)−2c2F(x)x−2c2xF(x)−3F(c2)x2−3x2F(c2)+3cF(c)x2+3cx2F(c)+4xF(c2)x−2cxF(c)x.0=2{c}^{2}F\left({x}^{2})-2{c}^{2}F\left(x)x-2{c}^{2}xF\left(x)-3F\left({c}^{2}){x}^{2}-3{x}^{2}F\left({c}^{2})+3cF\left(c){x}^{2}+3c{x}^{2}F\left(c)+4xF\left({c}^{2})x-2cxF\left(c)x.The aforementioned equation can be rewritten as follows: 0=2c2F(x2)−2c2F(x)x−2c2xF(x)−3F(c2)x2−3x2F(c2)+3cF(c)x2+3cx2F(c)+6xF(c2)x−2xF(c2)x−6cxF(c)x+4cxF(c)x.\begin{array}{rcl}0& =& 2{c}^{2}F\left({x}^{2})-2{c}^{2}F\left(x)x-2{c}^{2}xF\left(x)-3F\left({c}^{2}){x}^{2}-3{x}^{2}F\left({c}^{2})+3cF\left(c){x}^{2}+3c{x}^{2}F\left(c)+6xF\left({c}^{2})x\\ & & -2xF\left({c}^{2})x-6cxF\left(c)x+4cxF\left(c)x.\end{array}Using the aforementioned equation and (39), we obtain 0=2c2F(x2)−2c2F(x)x−2c2xF(x)+4cxF(c)x−2xF(c2)x.0=2{c}^{2}F\left({x}^{2})-2{c}^{2}F\left(x)x-2{c}^{2}xF\left(x)+4cxF\left(c)x-2xF\left({c}^{2})x.The aforementioned equation can now be rewritten as follows: c2F(x2)=c2F(x)x+c2xF(x)+xF(c2)x−2cxF(c)x\hspace{-25.15em}{c}^{2}F\left({x}^{2})={c}^{2}F\left(x)x+{c}^{2}xF\left(x)+xF\left({c}^{2})x-2cxF\left(c)xand 4c2F(x2)=4c2F(x)x+4c2xF(x)−2x(2F(c2)−4cF(c))x.\hspace{-25.1em}4{c}^{2}F\left({x}^{2})=4{c}^{2}F\left(x)x+4{c}^{2}xF\left(x)-2x\left(2F\left({c}^{2})-4cF\left(c))x.Setting 2F(c2)−4cF(c)=q2F\left({c}^{2})-4cF\left(c)=q, we arrive at (40)4c2F(x2)=4c2F(x)x+4c2xF(x)−2xqx.4{c}^{2}F\left({x}^{2})=4{c}^{2}F\left(x)x+4{c}^{2}xF\left(x)-2xqx.If q=0q=0than F(x)F\left(x)is derivation, on the other hand, if q≠0q\ne 0, we can conclude as follows. Let us now introduce the mapping D:R→RD:R\to Rby (41)D(x)=4F(x)−qx−xq.D\left(x)=4F\left(x)-qx-xq.Obviously, the mapping DDis additive. It is our aim to prove that DDis a Jordan derivation. Putting x2{x}^{2}for xxin the aforementioned relation, we obtain c2D(x2)=4c2F(x2)−c2qx2−c2x2q,{c}^{2}D\left({x}^{2})=4{c}^{2}F\left({x}^{2})-{c}^{2}q{x}^{2}-{c}^{2}{x}^{2}q,which gives, after considering the relation (40), the relation (42)c2D(x2)=4c2F(x)x+4c2xF(x)−2c2xqx−c2qx2−c2x2q.{c}^{2}D\left({x}^{2})=4{c}^{2}F\left(x)x+4{c}^{2}xF\left(x)-2{c}^{2}xqx-{c}^{2}q{x}^{2}-{c}^{2}{x}^{2}q.Right (left) multiplication of the relation (41) by xxgives, respectively, (43)D(x)x=4F(x)x−qx2−xqxD\left(x)x=4F\left(x)x-q{x}^{2}-xqxand (44)xD(x)=4xF(x)−xqx−x2q.xD\left(x)=4xF\left(x)-xqx-{x}^{2}q.The relations (42)–(44) imply that the additive mapping DDsatisfies the relation c2D(x2)=c2D(x)x+c2xD(x){c}^{2}D\left({x}^{2})={c}^{2}D\left(x)x+{c}^{2}xD\left(x)for all x∈Rx\in R, whence it follows D(x2)=D(x)x+xD(x)D\left({x}^{2})=D\left(x)x+xD\left(x)for all x∈Rx\in R. In other words, DDis a Jordan derivation on RR. According to Herstein theorem, one can conclude that DDis a derivation, which completes the proof of the theorem.□ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

# On certain functional equation in prime rings

, Volume 20 (1): 13 – Jan 1, 2022
13 pages

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de Gruyter
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© 2022 Maja Fošner et al., published by De Gruyter
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2391-5455
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2391-5455
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10.1515/math-2022-0002
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### Abstract

IntroductionThroughout, RRwill represent an associative ring with center Z(R)Z\left(R). Given an integer n>1n\gt 1, a ring RRis said to be nn-torsion free, if for x∈Rx\in R, nx=0nx=0implies x=0x=0. The commutator xy−yxxy-yxwill be denoted by [x,y]{[}x,y]. A ring RRis prime if for a,b∈Ra,b\in R, aRb=(0)aRb=\left(0)implies that either a=0a=0or b=0b=0and is semiprime in case aRa=(0)aRa=\left(0)implies a=0a=0. We denote by Qmr,Qs,{Q}_{mr},{Q}_{s},and CCthe maximal Martindale right ring of quotients, symmetric Martindale ring of quotients, and extended centroid of a semiprime ring RR, respectively (see [1], Chapter 2). An additive mapping D:R→RD:R\to Ris called a derivation if D(xy)=D(x)y+xD(y)D\left(xy)=D\left(x)y+xD(y)holds for all pairs x,y∈Rx,y\in Rand is called a Jordan derivation in case D(x2)=D(x)x+xD(x)D\left({x}^{2})=D\left(x)x+xD\left(x)is fulfilled for all x∈Rx\in R. A derivation D:R→RD:R\to Ris inner in case DDis of the form D(x)=[a,x]D\left(x)={[}a,x]for all x∈Rx\in Rand some fixed a∈Ra\in R. Every derivation is Jordan derivation. The converse is in general not true. A classical result of Herstein [2] asserts that any Jordan derivation on a prime ring with characteristic different from two is a derivation. A brief proof of Herstein theorem can be found in [3]. In [4], one can find a generalization of Herstein theorem. Cusack [5] generalized Herstein theorem to 2-torsion free semiprime rings (see [6] for an alternative proof). Herstein theorem has been fairly generalized by Beidar et al. [7]. For results related to Herstein theorem, we refer to [8,9, 10,11]. We proceed with the following result proved by Brešar [12] (see [13] for a generalization).Theorem 1Let R be a 2-torsion free semiprime ring and let D:R→RD:R\to Rbe an additive mapping satisfying the relation.(1)D(xyx)=D(x)yx+xD(y)x+xyD(x)D\left(xyx)=D\left(x)yx+xD(y)x+xyD\left(x)for all pairs x,y∈Rx,y\in R. In this case, DDis a derivation.An additive mapping satisfying the relation (1) on an arbitrary ring is called a Jordan triple derivation. It is easy to prove that any Jordan derivation on a 2-torsion free ring is a Jordan triple derivation, which means that Theorem 1 generalizes Cusack’s generalization of Herstein theorem.Motivated by Theorem 1, Vukman et al. [14] have proved the following result (see [15] for a generalization).Theorem 2Let R be a 2-torsion free semiprime ring and let F:R→RF:R\to Rbe an additive mapping satisfying the relation(2)T(xyx)=T(x)yx−xT(y)x+xyT(x)T\left(xyx)=T\left(x)yx-xT(y)x+xyT\left(x)for all pairs x,y∈Rx,y\in R. In this case, FFis of the form 2T(x)=qx+xq2T\left(x)=qx+xq, where q∈Qs(R)q\in {Q}_{s}\left(R)is some fixed element.We proceed with the following functional equation: (3)F(xyx)=F(xy)x−xF(y)x+xF(yx),F\left(xyx)=F\left(xy)x-xF(y)x+xF(yx),which appears naturally in the proof of Theorem 2 in [16]. One can easily prove that in case we have an additive mapping F:R→R,F:R\to R,where RRis 2-torsion free semiprime ring, satisfying the relation (3) for all pairs x,y∈Rx,y\in R, then FFis of the form 2F(x)=D(x)+ax+xa,2F\left(x)=D\left(x)+ax+xa,where D:R→RD:R\to Ris a derivation and a∈Ra\in Rsome fixed element (see [16] for the details). In [16], one can find the following conjecture.Conjecture 3Let R be a 2-torsion free semiprime ring and let F:R→RF:R\to Rbe an additive mapping satisfying the relation (3) for all pairs x,y∈Rx,y\in R. In this case, F is of the form 2F(x)=D(x)+qx+xq2F\left(x)=D\left(x)+qx+xqfor all x∈Rx\in R, where D:R→RD:R\to Ris a derivation and q∈Qs(R)q\in {Q}_{s}\left(R)some fixed element.By our knowledge, the aforementioned conjecture is still an open question. The substitution y=xy=xin (1), (2), and (3) gives (4)D(x3)=D(x)x2+xD(x)x+x2D(x),\hspace{0.45em}D\left({x}^{3})=D\left(x){x}^{2}+xD\left(x)x+{x}^{2}D\left(x),(5)F(x3)=F(x)x2−xF(x)x+x2F(x)F\left({x}^{3})=F\left(x){x}^{2}-xF\left(x)x+{x}^{2}F\left(x)and (6)F(x3)=F(x2)x−xF(x)x+xF(x2).F\left({x}^{3})=F\left({x}^{2})x-xF\left(x)x+xF\left({x}^{2}).The relation (4) has been considered in [7] (actually, much more general situation has been considered). A result related to (5) can be found in [15]. It is our aim in this paper to prove the following result, which is related to the aforementioned conjecture.Theorem 4Let RRbe a prime ring of characteristic different from two and three, and let F:R→RF:R\to Rbe an additive mapping satisfying the relation(7)F(x3)=F(x2)x−xF(x)x+xF(x2)F\left({x}^{3})=F\left({x}^{2})x-xF\left(x)x+xF\left({x}^{2})for all x∈Rx\in R. In this case, F is of the form 4F(x)=D(x)+qx+xq4F\left(x)=D\left(x)+qx+xq, where D:R→RD:R\to Ris a derivation, and q∈Qs(R)q\in {Q}_{s}\left(R)is some fixed element.Main resultsAs the main tool in this paper, we use the theory of functional identities (Brešar-Beidar-Chebotar theory). The theory of functional identities considers set-theoretic maps on rings that satisfy some identical relations. When treating such relations, one usually concludes that the form of the mappings involved can be described, unless the ring is very special. We refer the reader to [17] for the introductory account on the theory of functional identities, where Brešar presents this theory and its applications to a wider audience and to [18] for the full treatment of this theory.Let RRbe an algebra over a commutative ring ϕ\phi and let (8)p(x1,x2,x3)=∑π∈S3xπ(1)xπ(2)xπ(3)p\left({x}_{1},{x}_{2},{x}_{3})=\sum _{\pi \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}{x}_{\pi \left(2)}{x}_{\pi \left(3)}be a fixed multilinear polynomial in noncommuting indeterminates xi{x}_{i}over ϕ\phi . Here, S3{{\mathbb{S}}}_{3}stands for the symmetric group of order 3. Let ℒ{\mathcal{ {\mathcal L} }}be a subset of RRclosed under pp, i.e., p(x¯3)∈ℒp\left({\bar{x}}_{3})\in {\mathcal{ {\mathcal L} }}for all x1,x2,x3∈ℒ{x}_{1},{x}_{2},{x}_{3}\in {\mathcal{ {\mathcal L} }}, where x¯3=(x1,x2,x3){\bar{x}}_{3}=\left({x}_{1},{x}_{2},{x}_{3}). We shall consider a mapping D:ℒ→RD:{\mathcal{ {\mathcal L} }}\to Rsatisfying (9)F(p(x¯3))=∑π∈S3(F(xπ(1)xπ(2))xπ(3)−xπ(1)F(xπ(2))xπ(3)+xπ(1)F(xπ(2)xπ(3)))F\left(p\left({\bar{x}}_{3}))=\sum _{\pi \in {{\mathbb{S}}}_{3}}\left(F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){x}_{\pi \left(3)}-{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}+{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)}))for all x1,x2,x3∈ℒ{x}_{1},{x}_{2},{x}_{3}\in {\mathcal{ {\mathcal L} }}. Let us mention that the idea of considering the expression [p(x¯3),p(y¯3)]\left[p\left({\overline{x}}_{3}),p({\overline{y}}_{3})]in its proof is taken from [19]. For the proof of Theorem 4, we need Theorem 5, which might be of independent interest.Theorem 5Let ℒ{\mathcal{ {\mathcal L} }}be a 6-free Lie subring of R closed under p. If T:ℒ→RT:{\mathcal{ {\mathcal L} }}\to Ris an additive mapping satisfying (9), then there exists q∈Rq\in Rsuch that 4F(x)=D(x)+xq+qx4F\left(x)=D\left(x)+xq+qxfor all x∈ℒx\in {\mathcal{ {\mathcal L} }}.ProofFor any a∈Ra\in Rand x¯3∈ℒ3{\bar{x}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}, we have [p(x¯3),a]=p([x1,a],x2,x3)+p(x1,[x2,a],x3)+p(x1,x2,[x3,a]).\left[p\left({\bar{x}}_{3}),a]=p\left(\left[{x}_{1},a],{x}_{2},{x}_{3})+p\left({x}_{1},\left[{x}_{2},a],{x}_{3})+p\left({x}_{1},{x}_{2},\left[{x}_{3},a]).Thus, (10)F[p(x¯3),a]=F(p([x1,a],x2,x3))+F(p(x1,[x2,a],x3))+F(p(x1,x2,[x3,a])).F\left[p\left({\bar{x}}_{3}),a]=F\left(p\left(\left[{x}_{1},a],{x}_{2},{x}_{3}))+F\left(p\left({x}_{1},\left[{x}_{2},a],{x}_{3}))+F\left(p\left({x}_{1},{x}_{2},\left[{x}_{3},a])).By using (10), it follows that F[p(x¯3),a]=∑π∈S3F([xπ(1),a]xπ(2))xπ(3)−∑π∈S3[xπ(1),a]F(xπ(2))xπ(3)+∑π∈S3[xπ(1),a]F(xπ(2)xπ(3))+∑π∈S3F(xπ(1)[xπ(2),a])xπ(3)−∑π∈S3xπ(1)F([xπ(2),a])xπ(3)+∑π∈S3xπ(1)F([xπ(2),a]xπ(3))+∑π∈S3F(xπ(1)xπ(2))[xπ(3),a]−∑π∈S3xπ(1)F(xπ(2))[xπ(3),a]+∑π∈S3xπ(1)F(xπ(2)[xπ(3),a])=∑π∈S3F([xπ(1)xπ(2),a])xπ(3)−∑π∈S3[xπ(1),a]F(xπ(2))xπ(3)+∑π∈S3[xπ(1),a]F(xπ(2)xπ(3))−∑π∈S3xπ(1)F([xπ(2),a])xπ(3)+∑π∈S3xπ(1)F([xπ(2)xπ(3),a])+∑π∈S3F(xπ(1)xπ(2))[xπ(3),a]−∑π∈S3xπ(1)F(xπ(2))[xπ(3),a].\begin{array}{rcl}F\left[p\left({\bar{x}}_{3}),a]& =& \displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}F\left(\left[{x}_{\pi \left(1)},a]{x}_{\pi \left(2)}){x}_{\pi \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)},a]F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)},a]F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)})\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}F\left({x}_{\pi \left(1)}\left[{x}_{\pi \left(2)},a]){x}_{\pi \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)},a]){x}_{\pi \left(3)}+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)},a]{x}_{\pi \left(3)})\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},a]-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},a]+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}\left[{x}_{\pi \left(3)},a])\\ & =& \displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},a]){x}_{\pi \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)},a]F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)},a]F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)})\\ & & -\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)},a]){x}_{\pi \left(3)}+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},a])+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},a]\\ & & -\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},a].\end{array}In particular, (11)F[p(x¯3),p(y¯3)]=∑π∈S3F([xπ(1)xπ(2),p(y¯3)])xπ(3)−∑π∈S3[xπ(1),p(y¯3)]F(xπ(2))xπ(3)+∑π∈S3[xπ(1),p(y¯3)]F(xπ(2)xπ(3))−∑π∈S3xπ(1)F([xπ(2),p(y¯3)])xπ(3)+∑π∈S3xπ(1)F([xπ(2)xπ(3),p(y¯3)])+∑π∈S3F(xπ(1)xπ(2))[xπ(3),p(y¯3)]−∑π∈S3xπ(1)F(xπ(2))[xπ(3),p(y¯3)]\hspace{-38.6em}\begin{array}{rcl}F\left[p\left({\bar{x}}_{3}),p({\bar{y}}_{3})]& =& \displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},p({\bar{y}}_{3})]){x}_{\pi \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)},p({\bar{y}}_{3})]F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)},p({\bar{y}}_{3})]F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)})-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)},p({\bar{y}}_{3})]){x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},p({\bar{y}}_{3})])+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},p({\bar{y}}_{3})]\\ & & -\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},p({\bar{y}}_{3})]\end{array}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}. For i=1,2i=1,2, we have (12)F[xπ(i)xπ(i+1),p(y¯3)]=−F[p(y¯3),xπ(i)xπ(i+1)]=∑σ∈S3F([xπ(i)xπ(i+1),yσ(1)yσ(2)])yσ(3)−∑σ∈S3[xπ(i)xπ(i+1),yσ(1)]F(yσ(2))yσ(3)+∑σ∈S3[xπ(i)xπ(i+1),yσ(1)]F(yσ(2)yσ(3))−∑σ∈S3yσ(1)F([xπ(i)xπ(i+1),yσ(2)])yσ(3)\hspace{-38.7em}\begin{array}{rcl}F\left[{x}_{\pi \left(i)}{x}_{\pi \left(i+1)},p({\bar{y}}_{3})]& =& -F\left[p({\bar{y}}_{3}),{x}_{\pi \left(i)}{x}_{\pi \left(i+1)}]\\ & =& \displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F\left(\left[{x}_{\pi \left(i)}{x}_{\pi \left(i+1)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}-\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(i)}{x}_{\pi \left(i+1)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(i)}{x}_{\pi \left(i+1)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)})-\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(i)}{x}_{\pi \left(i+1)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}\end{array}(12)+∑σ∈S3yσ(1)F([xπ(i)xπ(i+1),yσ(2)yσ(3)])+∑σ∈S3F(yσ(1)yσ(2))[xπ(i)xπ(i+1),yσ(3)]−∑σ∈S3yσ(1)F(yσ(2))[xπ(i)xπ(i+1),yσ(3)]\begin{array}{rcl}& & +\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(i)}{x}_{\pi \left(i+1)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}])+\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)})\left[{x}_{\pi \left(i)}{x}_{\pi \left(i+1)},{y}_{\sigma \left(3)}]\\ & & -\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)})\left[{x}_{\pi \left(i)}{x}_{\pi \left(i+1)},{y}_{\sigma \left(3)}]\end{array}and F[xπ(2),p(y¯3)]=−F[p(y¯3),xπ(2)]=∑σ∈S3F([xπ(2),yσ(1)yσ(2)])yσ(3)−∑σ∈S3[xπ(2),yσ(1)]F(yσ(2))yσ(3)+∑σ∈S3[xπ(2),yσ(1)]F(yσ(2)yσ(3))−∑σ∈S3yσ(1)F([xπ(2),yσ(2)])yσ(3)+∑σ∈S3yσ(1)F([xπ(2),yσ(2)yσ(3)])+∑σ∈S3F(yσ(1)yσ(2))[xπ(2),yσ(3)]−∑σ∈S3yσ(1)F(yσ(2))[xπ(2),yσ(3)]\hspace{-40.65em}\begin{array}{rcl}F\left[{x}_{\pi \left(2)},p({\bar{y}}_{3})]& =& -F\left[p({\bar{y}}_{3}),{x}_{\pi \left(2)}]\\ & =& \displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}-\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)})-\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}])+\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)})\left[{x}_{\pi \left(2)},{y}_{\sigma \left(3)}]\\ & & -\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)})\left[{x}_{\pi \left(2)},{y}_{\sigma \left(3)}]\end{array}for all y¯3∈ℒ3{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}. Therefore, (11) can be written as follows: F[p(x¯3),p(y¯3)]=∑π∈S3∑σ∈S3F([xπ(1)xπ(2),yσ(1)yσ(2)])yσ(3)xπ(3)−∑π∈S3∑σ∈S3[xπ(1)xπ(2),yσ(1)]F(yσ(2))yσ(3)xπ(3)+∑π∈S3∑σ∈S3[xπ(1)xπ(2),yσ(1)]F(yσ(2)yσ(3))xπ(3)−∑π∈S3∑σ∈S3yσ(1)F([xπ(1)xπ(2),yσ(2)])yσ(3)xπ(3)+∑π∈S3∑σ∈S3yσ(1)F([xπ(1)xπ(2),yσ(2)yσ(3)])xπ(3)+∑π∈S3∑σ∈S3F(yσ(1)yσ(2))[xπ(1)xπ(2),yσ(3)]xπ(3)−∑π∈S3∑σ∈S3yσ(1)F(yσ(2))[xπ(1)xπ(2),yσ(3)]xπ(3)−∑π∈S3∑σ∈S3[xπ(1),yσ(1)yσ(2)yσ(3)]F(xπ(2))xπ(3)+∑π∈S3∑σ∈S3[xπ(1),yσ(1)yσ(2)yσ(3)]F(xπ(2)xπ(3))−∑π∈S3∑σ∈S3xπ(1)F([xπ(2),yσ(1)yσ(2)])yσ(3)xπ(3)+∑π∈S3∑σ∈S3xπ(1)[xπ(2),yσ(1)]F(yσ(2))yσ(3)xπ(3)−∑π∈S3∑σ∈S3xπ(1)[xπ(2),yσ(1)]F(yσ(2)yσ(3))xπ(3)+∑π∈S3∑σ∈S3xπ(1)yσ(1)F([xπ(2),yσ(2)])yσ(3)xπ(3)−∑π∈S3∑σ∈S3xπ(1)yσ(1)F([xπ(2),yσ(2)yσ(3)])xπ(3)−∑π∈S3∑σ∈S3xπ(1)F(yσ(1)yσ(2))[xπ(2),yσ(3)]xπ(3)+∑π∈S3∑σ∈S3xπ(1)yσ(1)F(yσ(2))[xπ(2),yσ(3)]xπ(3)+∑π∈S3∑σ∈S3xπ(1)F([xπ(2)xπ(3),yσ(1)yσ(2)])yσ(3)−∑π∈S3∑σ∈S3xπ(1)[xπ(2)xπ(3),yσ(1)]F(yσ(2))yσ(3)+∑π∈S3∑σ∈S3xπ(1)[xπ(2)xπ(3),yσ(1)]F(yσ(2)yσ(3))−∑π∈S3∑σ∈S3xπ(1)yσ(1)F([xπ(2)xπ(3),yσ(2)])yσ(3)+∑π∈S3∑σ∈S3xπ(1)yσ(1)F([xπ(2)xπ(3),yσ(2)yσ(3)])+∑π∈S3∑σ∈S3xπ(1)F(yσ(1)yσ(2))[xπ(2)xπ(3),yσ(3)]−∑π∈S3∑σ∈S3xπ(1)yσ(1)F(yσ(2))[xπ(2)xπ(3),yσ(3)]+∑π∈S3∑σ∈S3F(xπ(1)xπ(2))[xπ(3),yσ(1)yσ(2)yσ(3)]−∑π∈S3∑σ∈S3xπ(1)F(xπ(2))[xπ(3),yσ(1)yσ(2)yσ(3)]\begin{array}{rcl}F\left[p\left({\bar{x}}_{3}),p({\bar{y}}_{3})]& =& \displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)}){x}_{\pi \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]){x}_{\pi \left(3)}+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)})\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(3)}]{x}_{\pi \left(3)}\\ & & -\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)})\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(3)}]{x}_{\pi \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)})-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)}){x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]){x}_{\pi \left(3)}\\ & & -\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)})\left[{x}_{\pi \left(2)},{y}_{\sigma \left(3)}]{x}_{\pi \left(3)}+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)})\left[{x}_{\pi \left(2)},{y}_{\sigma \left(3)}]{x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)})-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}])+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)})\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(3)}]\\ & & -\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)})\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(3)}]+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]\\ & & -\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]\end{array}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}. On the other hand, by using [p(x¯3),p(y¯3)]=−[p(y¯3),p(x¯3)]\left[p\left({\bar{x}}_{3}),p({\bar{y}}_{3})]=-\left[p({\bar{y}}_{3}),p\left({\bar{x}}_{3})], we obtain from aforementioned identity F[p(x¯3),p(y¯3)]=∑π∈S3∑σ∈S3F([xπ(1)xπ(2),yσ(1)yσ(2)])xπ(3)yσ(3)−∑π∈S3∑σ∈S3[xπ(1),yσ(1)yσ(2)]F(xπ(2))xπ(3)yσ(3)+∑π∈S3∑σ∈S3[xπ(1),yσ(1)yσ(2)]F(xπ(2)xπ(3))yσ(3)−∑π∈S3∑σ∈S3xπ(1)F([xπ(2),yσ(1)yσ(2)])xπ(3)yσ(3)+∑π∈S3∑σ∈S3xπ(1)F([xπ(2)xπ(3),yσ(1)yσ(2)])yσ(3)+∑π∈S3∑σ∈S3F(xπ(1)xπ(2))[xπ(3),yσ(1)yσ(2)]yσ(3)\begin{array}{rcl}F\left[p\left({\bar{x}}_{3}),p({\bar{y}}_{3})]& =& \displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)}){y}_{\sigma \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]{y}_{\sigma \left(3)}\end{array}−∑π∈S3∑σ∈S3xπ(1)F(xπ(2))[xπ(3),yσ(1)yσ(2)]yσ(3)−∑π∈S3∑σ∈S3[xπ(1)xπ(2)xπ(3),yσ(1)]F(yσ(2))yσ(3)+∑π∈S3∑σ∈S3[xπ(1)xπ(2)xπ(3),yσ(1)]F(yσ(2)yσ(3))−∑π∈S3∑σ∈S3yσ(1)F([xπ(1)xπ(2),yσ(2)])xπ(3)yσ(3)+∑π∈S3∑σ∈S3yσ(1)[xπ(1),yσ(2)]F(xπ(2))xπ(3)yσ(3)−∑π∈S3∑σ∈S3yσ(1)[xπ(1),yσ(2)]F(xπ(2)xπ(3))yσ(3)+∑π∈S3∑σ∈S3yσ(1)xπ(1)F([xπ(2),yσ(2)])xπ(3)yσ(3)−∑π∈S3∑σ∈S3yσ(1)xπ(1)F([xπ(2)xπ(3),yσ(2)])yσ(3)−∑π∈S3∑σ∈S3yσ(1)F(xπ(1)xπ(2))[xπ(3),yσ(2)]yσ(3)+∑π∈S3∑σ∈S3yσ(1)xπ(1)F(xπ(2))[xπ(3),yσ(2)]yσ(3)+∑π∈S3∑σ∈S3yσ(1)F([xπ(1)xπ(2),yσ(2)yσ(3)])xπ(3)−∑π∈S3∑σ∈S3yσ(1)[xπ(1),yσ(2)yσ(3)]F(xπ(2))xπ(3)+∑π∈S3∑σ∈S3yσ(1)[xπ(1),yσ(2)yσ(3)]F(xπ(2)xπ(3))−∑π∈S3∑σ∈S3yσ(1)xπ(1)F([xπ(2),yσ(2)yσ(3)])xπ(3)+∑π∈S3∑σ∈S3yσ(1)xπ(1)F([xπ(2)xπ(3),yσ(2)yσ(3)])+∑π∈S3∑σ∈S3yσ(1)F(xπ(1)xπ(2))[xπ(3),yσ(2)yσ(3)]−∑π∈S3∑σ∈S3yσ(1)xπ(1)F(xπ(2))[xπ(3),yσ(2)yσ(3)]+∑π∈S3∑σ∈S3F(yσ(1)yσ(2))[xπ(1)xπ(2)xπ(3),yσ(3)]−∑π∈S3∑σ∈S3yσ(1)F(yσ(2))[xπ(1)xπ(2)xπ(3),yσ(3)]\begin{array}{rcl}& & -\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]{y}_{\sigma \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(1)}]F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)})-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}\left[{x}_{\pi \left(1)},{y}_{\sigma \left(2)}]F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}\left[{x}_{\pi \left(1)},{y}_{\sigma \left(2)}]F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)}){y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}\\ & & -\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},{y}_{\sigma \left(2)}]{y}_{\sigma \left(3)}+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},{y}_{\sigma \left(2)}]{y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]){x}_{\pi \left(3)}-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}\left[{x}_{\pi \left(1)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}\left[{x}_{\pi \left(1)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)})-\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]){x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}])+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]\\ & & -\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)})\left[{x}_{\pi \left(3)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)})\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(3)}]\\ & & -\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)})\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(3)}]\end{array}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}. By comparing so obtained identities, we arrive at (13)0=∑π∈S3∑σ∈S3F([xπ(1)xπ(2),yσ(1)yσ(2)])yσ(3)xπ(3)−F(xπ(1)xπ(2))yσ(1)yσ(2)yσ(3)xπ(3)+F(yσ(1)yσ(2))xπ(1)xπ(2)yσ(3)xπ(3)−yσ(1)F(yσ(2))xπ(1)xπ(2)yσ(3)xπ(3)−xπ(1)F(yσ(1)yσ(2))xπ(2)yσ(3)xπ(3)−yσ(1)xπ(1)F([yσ(2)yσ(3),xπ(2)])xπ(3)+xπ(1)yσ(1)F(yσ(2))xπ(2)yσ(3)xπ(3)+xπ(1)F(xπ(2))yσ(1)yσ(2)yσ(3)xπ(3)+yσ(1)F(xπ(1)xπ(2))yσ(2)yσ(3)xπ(3)−yσ(1)xπ(1)F(xπ(2))yσ(2)yσ(3)xπ(3)+∑π∈S3∑σ∈S3F([yσ(1)yσ(2),xπ(1)xπ(2)])xπ(3)yσ(3)−F(yσ(1)yσ(2))xπ(1)xπ(2)xπ(3)yσ(3)+F(xπ(1)xπ(2))yσ(1)yσ(2)xπ(3)yσ(3)−xπ(1)F(xπ(2))yσ(1)yσ(2)xπ(3)yσ(3)−yσ(1)F(xπ(1)xπ(2))yσ(2)xπ(3)yσ(3)−xπ(1)yσ(1)F([xπ(2)xπ(3),yσ(2)])yσ(3)+yσ(1)xπ(1)F(xπ(2))yσ(2)xπ(3)yσ(3)+yσ(1)F(yσ(2))xπ(1)xπ(2)xπ(3)yσ(3)+xπ(1)F(yσ(1)yσ(2))xπ(2)xπ(3)yσ(3)−xπ(1)yσ(1)F(yσ(2))xπ(2)xπ(3)yσ(3)+∑π∈S3∑σ∈S3xπ(1)yσ(1)F([xπ(2)xπ(3),yσ(2)yσ(3)])+xπ(1)yσ(1)xπ(2)xπ(3)F(yσ(2))yσ(3)−xπ(1)yσ(1)xπ(2)xπ(3)F(yσ(2)yσ(3))+xπ(1)yσ(1)yσ(2)yσ(3)F(xπ(2)xπ(3))−xπ(1)yσ(1)xπ(2)F(yσ(2))yσ(3)xπ(3)+xπ(1)yσ(1)xπ(2)F(yσ(2)yσ(3))xπ(3)−xπ(1)yσ(1)yσ(2)yσ(3)F(xπ(2))xπ(3)−xπ(1)F([xπ(2),yσ(1)yσ(2)])yσ(3)xπ(3)+xπ(1)yσ(1)F([xπ(2),yσ(2)])yσ(3)xπ(3)−xπ(1)yσ(1)F([xπ(2),yσ(2)yσ(3)])xπ(3)−xπ(1)F([yσ(1)yσ(2),xπ(2)])xπ(3)yσ(3)+xπ(1)yσ(1)yσ(2)F(xπ(2))xπ(3)yσ(3)−xπ(1)yσ(1)yσ(2)F(xπ(2)xπ(3))yσ(3)+∑π∈S3∑σ∈S3yσ(1)xπ(1)F([yσ(2)yσ(3),xπ(2)xπ(3)])+yσ(1)xπ(1)yσ(2)yσ(3)F(xπ(2))xπ(3)−yσ(1)xπ(1)yσ(2)yσ(3)F(xπ(2)xπ(3))+yσ(1)xπ(1)xπ(2)xπ(3)F(yσ(2)yσ(3))−yσ(1)xπ(1)yσ(2)F(xπ(2))xπ(3)yσ(3)+yσ(1)xπ(1)yσ(2)F(xπ(2)xπ(3))yσ(3)−yσ(1)xπ(1)xπ(2)xπ(3)F(yσ(2))yσ(3)\hspace{-36.15em}\begin{array}{rcl}0& =& \displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}-F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){y}_{\sigma \left(1)}{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(1)}{x}_{\pi \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}-{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(1)}{x}_{\pi \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & -{x}_{\pi \left(1)}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}-{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(2)}{y}_{\sigma \left(3)},{x}_{\pi \left(2)}]){x}_{\pi \left(3)}\\ & & +{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){y}_{\sigma \left(1)}{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +{y}_{\sigma \left(1)}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){y}_{\sigma \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}-{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){y}_{\sigma \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}F\left([{y}_{\sigma \left(1)}{y}_{\sigma \left(2)},{x}_{\pi \left(1)}{x}_{\pi \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}-F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(1)}{x}_{\pi \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & +F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){y}_{\sigma \left(1)}{y}_{\sigma \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){y}_{\sigma \left(1)}{y}_{\sigma \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & -{y}_{\sigma \left(1)}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){y}_{\sigma \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}\\ & & +{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){y}_{\sigma \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(1)}{x}_{\pi \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & +{x}_{\pi \left(1)}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}])\\ & & +{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{x}_{\pi \left(2)}{x}_{\pi \left(3)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}-{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{x}_{\pi \left(2)}{x}_{\pi \left(3)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)})\\ & & +{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)})-{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)}){x}_{\pi \left(3)}-{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}\\ & & -{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & -{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]){x}_{\pi \left(3)}-{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(1)}{y}_{\sigma \left(2)},{x}_{\pi \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & +{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)}){y}_{\sigma \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(2)}{y}_{\sigma \left(3)},{x}_{\pi \left(2)}{x}_{\pi \left(3)}])\\ & & +{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}-{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)})\\ & & +{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{x}_{\pi \left(2)}{x}_{\pi \left(3)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)})-{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & +{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)}){y}_{\sigma \left(3)}-{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{x}_{\pi \left(2)}{x}_{\pi \left(3)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}\end{array}(13)−yσ(1)F([yσ(2),xπ(1)xπ(2)])xπ(3)yσ(3)+yσ(1)xπ(1)F([yσ(2),xπ(2)])xπ(3)yσ(3)−yσ(1)xπ(1)F([yσ(2),xπ(2)xπ(3)])yσ(3)−yσ(1)F([xπ(1)xπ(2),yσ(2)])yσ(3)xπ(3)−yσ(1)xπ(1)xπ(2)F(yσ(2)yσ(3))xπ(3)+yσ(1)xπ(1)xπ(2)F(yσ(2))yσ(3)xπ(3)\begin{array}{rcl}& & -{y}_{\sigma \left(1)}F\left([{y}_{\sigma \left(2)},{x}_{\pi \left(1)}{x}_{\pi \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(2)},{x}_{\pi \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & -{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(2)},{x}_{\pi \left(2)}{x}_{\pi \left(3)}]){y}_{\sigma \left(3)}-{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & -{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)}){x}_{\pi \left(3)}+{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\end{array}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}. By using the theory of functional identities, we can conclude that (14)∑π∈S2∑σ∈S2F([xπ(1)xπ(2),yσ(1)yσ(2)])−F(xπ(1)xπ(2))yσ(1)yσ(2)+F(yσ(1)yσ(2))xπ(1)xπ(2)−yσ(1)F(yσ(2))xπ(1)xπ(2)+xπ(1)F(xπ(2))yσ(1)yσ(2)=x1p1(x2,y1,y2)+x2p2(x1,y1,y2)+y1p3(x1,x2,y2)+y2p4(x1,x2,y1)+λp(x1,x2,y1,y2)\begin{array}{l}\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{2}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{2}}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}])-F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){y}_{\sigma \left(1)}{y}_{\sigma \left(2)}+F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(1)}{x}_{\pi \left(2)}\\ -{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(1)}{x}_{\pi \left(2)}+{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){y}_{\sigma \left(1)}{y}_{\sigma \left(2)}\\ \hspace{1.0em}={x}_{1}{p}_{1}\left({x}_{2},{y}_{1},{y}_{2})+{x}_{2}{p}_{2}\left({x}_{1},{y}_{1},{y}_{2})+{y}_{1}{p}_{3}\left({x}_{1},{x}_{2},{y}_{2})+{y}_{2}{p}_{4}\left({x}_{1},{x}_{2},{y}_{1})+{\lambda }_{p}\left({x}_{1},{x}_{2},{y}_{1},{y}_{2})\end{array}and (15)∑π∈S2∑σ∈S2F([xπ(2)xπ(3),yσ(2)yσ(3)])−xπ(2)xπ(3)F(yσ(2)yσ(3))+yσ(2)yσ(3)F(xπ(2)xπ(3))+xπ(2)xπ(3)F(yσ(2))yσ(3)−yσ(2)yσ(3)F(xπ(2))xπ(3)=q1(x3,y2,y3)x2+q2(x2,y2,y3)x3+q3(x2,x3,y3)y2+q4(x2,x3,y2)y3+λq(x2,x3,y2,y3).\hspace{-45em}\begin{array}{l}\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{2}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{2}}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}])-{x}_{\pi \left(2)}{x}_{\pi \left(3)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)})+{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)})+{x}_{\pi \left(2)}{x}_{\pi \left(3)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}\hspace{1em}-{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}={q}_{1}\left({x}_{3},{y}_{2},{y}_{3}){x}_{2}+{q}_{2}\left({x}_{2},{y}_{2},{y}_{3}){x}_{3}+{q}_{3}\left({x}_{2},{x}_{3},{y}_{3}){y}_{2}+{q}_{4}\left({x}_{2},{x}_{3},{y}_{2}){y}_{3}+{\lambda }_{q}\left({x}_{2},{x}_{3},{y}_{2},{y}_{3}).\end{array}We also have ∑π∈S2∑σ∈S2F([yσ(1)yσ(2),xπ(1)xπ(2)])−F(yσ(1)yσ(2))xπ(1)xπ(2)+F(xπ(1)xπ(2))yσ(1)yσ(2)−xπ(1)F(xπ(2))yσ(1)yσ(2)+yσ(1)F(yσ(2))xπ(1)xπ(2)=x1p1′(x2,y1,y2)+x2p2′(x1,y1,y2)+y1p3′(x1,x2,y2)+y2p4′(x1,x2,y1)+λp′(x1,x2,y1,y2)\begin{array}{l}\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{2}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{2}}F\left([{y}_{\sigma \left(1)}{y}_{\sigma \left(2)},{x}_{\pi \left(1)}{x}_{\pi \left(2)}])-F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(1)}{x}_{\pi \left(2)}+F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){y}_{\sigma \left(1)}{y}_{\sigma \left(2)}\\ \hspace{1.0em}-{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){y}_{\sigma \left(1)}{y}_{\sigma \left(2)}+{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(1)}{x}_{\pi \left(2)}\\ \hspace{1.0em}={x}_{1}{p}_{1}^{^{\prime} }\left({x}_{2},{y}_{1},{y}_{2})+{x}_{2}{p}_{2}^{^{\prime} }\left({x}_{1},{y}_{1},{y}_{2})+{y}_{1}{p}_{3}^{^{\prime} }\left({x}_{1},{x}_{2},{y}_{2})+{y}_{2}{p}_{4}^{^{\prime} }\left({x}_{1},{x}_{2},{y}_{1})+{\lambda }_{p^{\prime} }\left({x}_{1},{x}_{2},{y}_{1},{y}_{2})\end{array}and ∑π∈S2∑σ∈S2F([yσ(2)yσ(3),xπ(2)xπ(3)])−yσ(2)yσ(3)F(xπ(2)xπ(3))+xπ(2)xπ(3)F(yσ(2)yσ(3))−xπ(2)xπ(3)F(yσ(2))yσ(3)+yσ(2)yσ(3)F(xπ(2))xπ(3)=q1′(x3,y2,y3)x2+q2′(x2,y2,y3)x3+q3′(x2,x3,y3)y2+q4′(x2,x3,y2)y3+λq′(x2,x3,y2,y3)\begin{array}{l}\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{2}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{2}}F\left([{y}_{\sigma \left(2)}{y}_{\sigma \left(3)},{x}_{\pi \left(2)}{x}_{\pi \left(3)}])-{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)})+{x}_{\pi \left(2)}{x}_{\pi \left(3)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)})\\ \hspace{1.0em}-{x}_{\pi \left(2)}{x}_{\pi \left(3)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}+{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}\\ \hspace{1.0em}={q}_{1}^{^{\prime} }\left({x}_{3},{y}_{2},{y}_{3}){x}_{2}+{q}_{2}^{^{\prime} }\left({x}_{2},{y}_{2},{y}_{3}){x}_{3}+{q}_{3}^{^{\prime} }\left({x}_{2},{x}_{3},{y}_{3}){y}_{2}+{q}_{4}^{^{\prime} }\left({x}_{2},{x}_{3},{y}_{2}){y}_{3}+{\lambda }_{q^{\prime} }\left({x}_{2},{x}_{3},{y}_{2},{y}_{3})\end{array}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}and pi,pi′,qi,qi′:ℒ3→R{p}_{i},{p}_{i}^{^{\prime} },{q}_{i},{q}_{i}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{3}\to R, i=1,2,3,4i=1,2,3,4and λp,λp′,λq,λq′:ℒ4→C(ℒ){\lambda }_{p},{\lambda }_{p^{\prime} },{\lambda }_{q},{\lambda }_{q^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{4}\to C\left({\mathcal{ {\mathcal L} }}). By comparing identities (14) and (15), we arrive at (16)∑π∈S2∑σ∈S2(−F(xπ(1)xπ(2))yσ(1)yσ(2)+F(yσ(1)yσ(2))xπ(1)xπ(2)−yσ(1)F(yσ(2))xπ(1)xπ(2)+xπ(1)F(xπ(2))yσ(1)yσ(2)+xπ(1)xπ(2)F(yσ(1)yσ(2))−yσ(1)yσ(2)F(xπ(1)xπ(2))+yσ(1)yσ(2)F(xπ(1))xπ(2)−xπ(1)xπ(2)F(yσ(1))yσ(2))=xπ(1)p1(xπ(2),yσ(1),yσ(2))+xπ(2)p2(xπ(1),yσ(1),yσ(2))+yσ(1)p3(xπ(1),xπ(2),yσ(2))+yσ(2)p4(xπ(1),xπ(2),yσ(1))+λ1(xπ(1),xπ(2),yσ(1),yσ(2))−q1(xπ(2),yσ(1),yσ(2))xπ(1)−q2(xπ(1),yσ(1),yσ(2))xπ(2)−q3(xπ(1),xπ(2),yσ(2))yσ(1)−q4(xπ(1),xπ(2),yσ(1))yσ(2)−μ1(xπ(1),xπ(2),yσ(1),yσ(2))\hspace{-25em}\begin{array}{l}\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{2}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{2}}(-F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){y}_{\sigma \left(1)}{y}_{\sigma \left(2)}+F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(1)}{x}_{\pi \left(2)}-{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(1)}{x}_{\pi \left(2)}+{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){y}_{\sigma \left(1)}{y}_{\sigma \left(2)}\\ \hspace{1.0em}+{x}_{\pi \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)})-{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)})+{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(1)}){x}_{\pi \left(2)}-{x}_{\pi \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(1)}){y}_{\sigma \left(2)})\\ \hspace{1.0em}={x}_{\pi \left(1)}{p}_{1}\left({x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)})+{x}_{\pi \left(2)}{p}_{2}\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)})+{y}_{\sigma \left(1)}{p}_{3}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(2)})\\ \hspace{2.0em}+{y}_{\sigma \left(2)}{p}_{4}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)})+{\lambda }_{1}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)})-{q}_{1}\left({x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){x}_{\pi \left(1)}\\ \hspace{2.0em}-{q}_{2}\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){x}_{\pi \left(2)}-{q}_{3}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(2)}){y}_{\sigma \left(1)}-{q}_{4}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)}){y}_{\sigma \left(2)}-{\mu }_{1}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)})\end{array}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}and pi,pi′,qi,qi′:ℒ3→R{p}_{i},{p}_{i}^{^{\prime} },{q}_{i},{q}_{i}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{3}\to R, i=1,2,3,4i=1,2,3,4and λp,λp′,λq,λq′:ℒ4→C(ℒ){\lambda }_{p},{\lambda }_{p^{\prime} },{\lambda }_{q},{\lambda }_{q^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{4}\to C\left({\mathcal{ {\mathcal L} }}). By using equation (16) and the theory of functional identities, it follows that ∑σ∈S2(F(yσ(1)yσ(2))xπ(1)−yσ(1)F(yσ(2))xπ(1))+q2(xπ(1),yσ(1),yσ(2))=xπ(1)m1(yσ(1),yσ(2))+yσ(1)m2(xπ(1),yσ(2))+yσ(2)m3(xπ(1),yσ(1))+λm(xπ(1),yσ(1),yσ(2))\begin{array}{l}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{2}}\left(F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(1)}-{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(1)})+{q}_{2}\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)})\\ \hspace{1.0em}={x}_{\pi \left(1)}{m}_{1}({y}_{\sigma \left(1)},{y}_{\sigma \left(2)})+{y}_{\sigma \left(1)}{m}_{2}\left({x}_{\pi \left(1)},{y}_{\sigma \left(2)})+{y}_{\sigma \left(2)}{m}_{3}\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)})+{\lambda }_{m}\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)})\end{array}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}, q2:ℒ3→R{q}_{2}:{{\mathcal{ {\mathcal L} }}}^{3}\to R, mi:ℒ2→R{m}_{i}:{{\mathcal{ {\mathcal L} }}}^{2}\to R, i=1,2,3i=1,2,3and λm:ℒ3→C(ℒ){\lambda }_{m}:{{\mathcal{ {\mathcal L} }}}^{3}\to C\left({\mathcal{ {\mathcal L} }}). Now setting xπ(1),yσ(1),yσ(2)=x{x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}=xin the aforementioned equation, we obtain (17)2F(x2)x−2xF(x)x+q2(x,x,x)=xm1(x,x)+xm2(x,x)+xm3(x,x)+λm(x,x,x)2F\left({x}^{2})x-2xF\left(x)x+{q}_{2}\left(x,x,x)=x{m}_{1}\left(x,x)+x{m}_{2}\left(x,x)+x{m}_{3}\left(x,x)+{\lambda }_{m}\left(x,x,x)for all x∈Rx\in R.On the other hand, using equation (16) and the theory of functional identities, we arrive at ∑σ∈S2(xπ(1)F(yσ(1)yσ(2))−xπ(1)F(yσ(1))yσ(2))−p2(xπ(1),yσ(1),yσ(2))=n1′(yσ(1),yσ(2))xπ(1)+n2′(xπ(1),yσ(2))yσ(1)+n3′(xπ(1),yσ(1))yσ(2)+λn′(xπ(1),yσ(1),yσ(2))\begin{array}{l}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{2}}\left({x}_{\pi \left(1)}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)})-{x}_{\pi \left(1)}F({y}_{\sigma \left(1)}){y}_{\sigma \left(2)})-{p}_{2}\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)})\\ \hspace{1.0em}={n}_{1}^{^{\prime} }({y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){x}_{\pi \left(1)}+{n}_{2}^{^{\prime} }\left({x}_{\pi \left(1)},{y}_{\sigma \left(2)}){y}_{\sigma \left(1)}+{n}_{3}^{^{\prime} }\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)}){y}_{\sigma \left(2)}+{\lambda }_{n}^{^{\prime} }\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)})\end{array}x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}, p2:ℒ3→R{p}_{2}:{{\mathcal{ {\mathcal L} }}}^{3}\to R, ni′:ℒ2→R{n}_{i}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{2}\to R, i=1,2,3i=1,2,3and λn′:ℒ3→C(ℒ){\lambda }_{n}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{3}\to C\left({\mathcal{ {\mathcal L} }}).Now setting xπ(1),yσ(1),yσ(2)=x{x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}=x, we obtain (18)2xF(x2)−2xF(x)x−p2(x,x,x)=n1′(x,x)x+n2′(x,x)x+n3′(x,x)x+λn′(x,x,x)2xF\left({x}^{2})-2xF\left(x)x-{p}_{2}\left(x,x,x)={n}_{1}^{^{\prime} }\left(x,x)x+{n}_{2}^{^{\prime} }\left(x,x)x+{n}_{3}^{^{\prime} }\left(x,x)x+{\lambda }_{n}^{^{\prime} }\left(x,x,x)for all x∈Rx\in R. Equation (13) can now be rewritten as follows: 0=∑π∈S3∑σ∈S3(xπ(1)p1(xπ(2),yσ(1),yσ(2))yσ(3)xπ(3)+xπ(2)p2(xπ(1),yσ(1),yσ(2))yσ(3)xπ(3)+yσ(1)p3(xπ(1),xπ(2),yσ(2))yσ(3)xπ(3)+yσ(2)p4(xπ(1),xπ(2),yσ(1))yσ(3)xπ(3)+λp(xπ(1),xπ(2),yσ(1),yσ(2))yσ(3)xπ(3)+xπ(1)yσ(1)F(yσ(2))xπ(2)yσ(3)xπ(3)−xπ(1)F(yσ(1)yσ(2))xπ(2)yσ(3)xπ(3)−yσ(1)xπ(1)F([yσ(2)yσ(3),xπ(2)])xπ(3)+yσ(1)F(xπ(1)xπ(2))yσ(2)yσ(3)xπ(3)−yσ(1)xπ(1)F(xπ(2))yσ(2)yσ(3)xπ(3)+∑π∈S3∑σ∈S3(xπ(1)p1′(xπ(2),yσ(1),yσ(2))xπ(3)yσ(3)+xπ(2)p2′(xπ(1),yσ(1),yσ(2))xπ(3)yσ(3)+yσ(1)p3′(xπ(1),xπ(2),yσ(2))xπ(3)yσ(3)+yσ(2)p4′(xπ(1),xπ(2),yσ(1))xπ(3)yσ(3))+λp′(xπ(1),xπ(2),yσ(1),yσ(2))xπ(3)yσ(3)+yσ(1)xπ(1)F(xπ(2))yσ(2)xπ(3)yσ(3)−yσ(1)F(xπ(1)xπ(2))yσ(2)xπ(3)yσ(3)−xπ(1)yσ(1)F([xπ(2)xπ(3),yσ(2)])yσ(3)+xπ(1)F(yσ(1)yσ(2))xπ(2)xπ(3)yσ(3)−xπ(1)yσ(1)F(yσ(2))xπ(2)xπ(3)yσ(3))+∑π∈S3∑σ∈S3(xπ(1)yσ(1)q1(xπ(3),yσ(2),yσ(3))xπ(2)+xπ(1)yσ(1)q2(xπ(2),yσ(2),yσ(3))xπ(3)+xπ(1)yσ(1)q3(xπ(2),xπ(3),yσ(3))yσ(2)+xπ(1)yσ(1)q4(xπ(2),xπ(3),yσ(2))yσ(3)+xπ(1)yσ(1)λq(xπ(2),xπ(3),yσ(2),yσ(3))−xπ(1)yσ(1)xπ(2)F(yσ(2))yσ(3)xπ(3)+xπ(1)yσ(1)xπ(2)F(yσ(2)yσ(3))xπ(3)−xπ(1)F([xπ(2),yσ(1)yσ(2)])yσ(3)xπ(3)+xπ(1)yσ(1)F([xπ(2),yσ(2)])yσ(3)xπ(3)−xπ(1)yσ(1)F([xπ(2),yσ(2)yσ(3)])xπ(3)−xπ(1)F([yσ(1)yσ(2),xπ(2)])xπ(3)yσ(3)+xπ(1)yσ(1)yσ(2)F(xπ(2))xπ(3)yσ(3)−xπ(1)yσ(1)yσ(2)F(xπ(2)xπ(3))yσ(3))+∑π∈S3∑σ∈S3(yσ(1)xπ(1)q1′(xπ(3),yσ(2),yσ(3))xπ(2)+yσ(1)xπ(1)q2′(xπ(2),yσ(2),yσ(3))xπ(3)+yσ(1)xπ(1)q3′(xπ(2),xπ(3),yσ(3))yσ(2)+yσ(1)xπ(1)q4′(xπ(2),xπ(3),yσ(2))yσ(3)+yσ(1)xπ(1)λq′(xπ(2),xπ(3),yσ(2),yσ(3))−yσ(1)xπ(1)yσ(2)F(xπ(2))xπ(3)yσ(3)+yσ(1)xπ(1)yσ(2)F(xπ(2)xπ(3))yσ(3)−yσ(1)F([yσ(2),xπ(1)xπ(2)])xπ(3)yσ(3)+yσ(1)xπ(1)F([yσ(2),xπ(2)])xπ(3)yσ(3)−yσ(1)xπ(1)F([yσ(2),xπ(2)xπ(3)])yσ(3)−yσ(1)F([xπ(1)xπ(2),yσ(2)])yσ(3)xπ(3)−yσ(1)xπ(1)xπ(2)F(yσ(2)yσ(3))xπ(3)+yσ(1)xπ(1)xπ(2)F(yσ(2))yσ(3)xπ(3))\hspace{-36.75em}\begin{array}{rcl}0& =& \displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}({x}_{\pi \left(1)}{p}_{1}\left({x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{x}_{\pi \left(2)}{p}_{2}\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +{y}_{\sigma \left(1)}{p}_{3}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{y}_{\sigma \left(2)}{p}_{4}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +{\lambda }_{p}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & -{x}_{\pi \left(1)}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}-{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(2)}{y}_{\sigma \left(3)},{x}_{\pi \left(2)}]){x}_{\pi \left(3)}\\ & & +{y}_{\sigma \left(1)}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){y}_{\sigma \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}-{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){y}_{\sigma \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}({x}_{\pi \left(1)}{p}_{1}^{^{\prime} }\left({x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{x}_{\pi \left(2)}{p}_{2}^{^{\prime} }\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & +{y}_{\sigma \left(1)}{p}_{3}^{^{\prime} }\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{y}_{\sigma \left(2)}{p}_{4}^{^{\prime} }\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)})\\ & & +{\lambda }_{p^{\prime} }\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){y}_{\sigma \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & -{y}_{\sigma \left(1)}F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){y}_{\sigma \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}\\ & & +{x}_{\pi \left(1)}F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)})\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}({x}_{\pi \left(1)}{y}_{\sigma \left(1)}{q}_{1}\left({x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)}){x}_{\pi \left(2)}\\ & & +{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{q}_{2}\left({x}_{\pi \left(2)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)}){x}_{\pi \left(3)}+{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{q}_{3}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(3)}){y}_{\sigma \left(2)}\\ & & +{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{q}_{4}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}+{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{\lambda }_{q}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)})\\ & & -{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)}){x}_{\pi \left(3)}\\ & & -{x}_{\pi \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & -{x}_{\pi \left(1)}{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]){x}_{\pi \left(3)}-{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(1)}{y}_{\sigma \left(2)},{x}_{\pi \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & +{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{x}_{\pi \left(1)}{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)}){y}_{\sigma \left(3)})\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}({y}_{\sigma \left(1)}{x}_{\pi \left(1)}{q}_{1}^{^{\prime} }\left({x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)}){x}_{\pi \left(2)}\\ & & +{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{q}_{2}^{^{\prime} }\left({x}_{\pi \left(2)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)}){x}_{\pi \left(3)}+{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{q}_{3}^{^{\prime} }\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(3)}){y}_{\sigma \left(2)}\\ & & +{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{q}_{4}^{^{\prime} }\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}+{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{\lambda }_{q^{\prime} }\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)})\\ & & -{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)}){y}_{\sigma \left(3)}\\ & & -{y}_{\sigma \left(1)}F\left([{y}_{\sigma \left(2)},{x}_{\pi \left(1)}{x}_{\pi \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(2)},{x}_{\pi \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & -{y}_{\sigma \left(1)}{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(2)},{x}_{\pi \left(2)}{x}_{\pi \left(3)}]){y}_{\sigma \left(3)}-{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & -{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)}){x}_{\pi \left(3)}+{y}_{\sigma \left(1)}{x}_{\pi \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)})\end{array}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}, pi,pi′,qi,qi′:ℒ3→R{p}_{i},{p}_{i}^{^{\prime} },{q}_{i},{q}_{i}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{3}\to R, i=1,2,3,4i=1,2,3,4and λp,λp′,λq,λq′:ℒ4→C(ℒ){\lambda }_{p},{\lambda }_{p^{\prime} },{\lambda }_{q},{\lambda }_{q^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{4}\to C\left({\mathcal{ {\mathcal L} }}). By using the theory of functional identities and exposing everything from the same side (left), the aforementioned equation can now be rewritten as follows: (19)0=∑π∈S3∑σ∈S3xπ(1)(p1(xπ(2),yσ(1),yσ(2))yσ(3)xπ(3)−F(yσ(1)yσ(2))xπ(2)yσ(3)xπ(3)+yσ(1)F(yσ(2))xπ(2)yσ(3)xπ(3)+p1′(xπ(2),yσ(1),yσ(2))xπ(3)yσ(3)−yσ(1)F([xπ(2)xπ(3),yσ(2)])yσ(3)+F(yσ(1)yσ(2))xπ(2)xπ(3)yσ(3)−yσ(1)F(yσ(2))xπ(2)xπ(3)yσ(3)+yσ(1)q1(xπ(3),yσ(2),yσ(3))xπ(2)\begin{array}{rcl}0& =& \displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(1)}({p}_{1}\left({x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}-F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{p}_{1}^{^{\prime} }\left({x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}\\ & & +F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{y}_{\sigma \left(1)}{q}_{1}\left({x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)}){x}_{\pi \left(2)}\end{array}(19)+yσ(1)q2(xπ(2),yσ(2),yσ(3))xπ(3)+yσ(1)q3(xπ(2),xπ(3),yσ(3))yσ(2)+yσ(1)q4(xπ(2),xπ(3),yσ(2))yσ(3)+yσ(1)λq(xπ(2),xπ(3),yσ(2),yσ(3))−yσ(1)xπ(2)F(yσ(2))yσ(3)xπ(3)+yσ(1)xπ(2)F(yσ(2)yσ(3))xπ(3)−F([xπ(2),yσ(1)yσ(2)])yσ(3)xπ(3)+yσ(1)F([xπ(2),yσ(2)])yσ(3)xπ(3)−yσ(1)F([xπ(2),yσ(2)yσ(3)])xπ(3)−F([yσ(1)yσ(2),xπ(2)])xπ(3)yσ(3)+yσ(1)yσ(2)F(xπ(2))xπ(3)yσ(3)−yσ(1)yσ(2)F(xπ(2)xπ(3))yσ(3))+∑π∈S3∑σ∈S3xπ(2)(p2(xπ(1),yσ(1),yσ(2))yσ(3)xπ(3)+p2′(xπ(1),yσ(1),yσ(2))xπ(3)yσ(3))+∑π∈S3∑σ∈S3xπ(3)(yσ(3)λp′(xπ(1),xπ(2),yσ(1),yσ(2)))+∑π∈S3∑σ∈S3yσ(1)(p3(xπ(1),xπ(2),yσ(2))yσ(3)xπ(3)−xπ(1)F([yσ(2)yσ(3),xπ(2)])xπ(3)+F(xπ(1)xπ(2))yσ(2)yσ(3)xπ(3)−xπ(1)F(xπ(2))yσ(2)yσ(3)xπ(3)+p3′(xπ(1),xπ(2),yσ(2))xπ(3)yσ(3)−F(xπ(1)xπ(2))yσ(2)xπ(3)yσ(3)+xπ(1)F(xπ(2))yσ(2)xπ(3)yσ(3)+xπ(1)q1′(xπ(3),yσ(2),yσ(3))xπ(2)+xπ(1)q2′(xπ(2),yσ(2),yσ(3))xπ(3)+xπ(1)q3′(xπ(2),xπ(3),yσ(3))yσ(2)+xπ(1)q4′(xπ(2),xπ(3),yσ(2))yσ(3)+xπ(1)λq′(xπ(2),xπ(3),yσ(2),yσ(3))−xπ(1)yσ(2)F(xπ(2))xπ(3)yσ(3)+xπ(1)yσ(2)F(xπ(2)xπ(3))yσ(3)−F([yσ(2),xπ(1)xπ(2)])xπ(3)yσ(3)+xπ(1)F([yσ(2),xπ(2)])xπ(3)yσ(3)−xπ(1)F([yσ(2),xπ(2)xπ(3)])yσ(3)−F([xπ(1)xπ(2),yσ(2)])yσ(3)xπ(3)−xπ(1)xπ(2)F(yσ(2)yσ(3))xπ(3)+xπ(1)xπ(2)F(yσ(2))yσ(3)xπ(3))+∑π∈S3∑σ∈S3yσ(2)(p4(xπ(1),xπ(2),yσ(1))yσ(3)xπ(3)+p4′(xπ(1),xπ(2),yσ(1))xπ(3)yσ(3))+∑π∈S3∑σ∈S3yσ(3)(xπ(3)λp(xπ(1),xπ(2),yσ(1),yσ(2)))\begin{array}{rcl}& & +{y}_{\sigma \left(1)}{q}_{2}\left({x}_{\pi \left(2)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)}){x}_{\pi \left(3)}+{y}_{\sigma \left(1)}{q}_{3}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(3)}){y}_{\sigma \left(2)}+{y}_{\sigma \left(1)}{q}_{4}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}\\ & & +{y}_{\sigma \left(1)}{\lambda }_{q}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)})-{y}_{\sigma \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{y}_{\sigma \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)}){x}_{\pi \left(3)}\\ & & -F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}-{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]){x}_{\pi \left(3)}\\ & & -F\left([{y}_{\sigma \left(1)}{y}_{\sigma \left(2)},{x}_{\pi \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)}){y}_{\sigma \left(3)})\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(2)}({p}_{2}\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{p}_{2}^{^{\prime} }\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)})\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{x}_{\pi \left(3)}({y}_{\sigma \left(3)}{\lambda }_{p^{\prime} }\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}))\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(1)}({p}_{3}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}-{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(2)}{y}_{\sigma \left(3)},{x}_{\pi \left(2)}]){x}_{\pi \left(3)}\\ & & +F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){y}_{\sigma \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}-{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){y}_{\sigma \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{p}_{3}^{^{\prime} }\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & -F\left({x}_{\pi \left(1)}{x}_{\pi \left(2)}){y}_{\sigma \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{x}_{\pi \left(1)}F\left({x}_{\pi \left(2)}){y}_{\sigma \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{x}_{\pi \left(1)}{q}_{1}^{^{\prime} }\left({x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)}){x}_{\pi \left(2)}\\ & & +{x}_{\pi \left(1)}{q}_{2}^{^{\prime} }\left({x}_{\pi \left(2)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)}){x}_{\pi \left(3)}+{x}_{\pi \left(1)}{q}_{3}^{^{\prime} }\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(3)}){y}_{\sigma \left(2)}+{x}_{\pi \left(1)}{q}_{4}^{^{\prime} }\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}\\ & & +{x}_{\pi \left(1)}{\lambda }_{q^{\prime} }\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)})-{x}_{\pi \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{x}_{\pi \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)}){y}_{\sigma \left(3)}\\ & & -F\left([{y}_{\sigma \left(2)},{x}_{\pi \left(1)}{x}_{\pi \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(2)},{x}_{\pi \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{x}_{\pi \left(1)}F\left([{y}_{\sigma \left(2)},{x}_{\pi \left(2)}{x}_{\pi \left(3)}]){y}_{\sigma \left(3)}\\ & & -F\left(\left[{x}_{\pi \left(1)}{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}-{x}_{\pi \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)}){x}_{\pi \left(3)}+{x}_{\pi \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)})\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(2)}({p}_{4}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{p}_{4}^{^{\prime} }\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)})\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{3}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}{y}_{\sigma \left(3)}\left({x}_{\pi \left(3)}{\lambda }_{p}\left({x}_{\pi \left(1)},{x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}))\end{array}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}, pi,pi′,qi,qi′:ℒ3→R{p}_{i},{p}_{i}^{^{\prime} },{q}_{i},{q}_{i}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{3}\to R, i=1,2,3,4i=1,2,3,4and λp,λp′,λq,λq′:ℒ4→C(ℒ){\lambda }_{p},{\lambda }_{p^{\prime} },{\lambda }_{q},{\lambda }_{q^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{4}\to C\left({\mathcal{ {\mathcal L} }}). Now by using the theory of functional identities and exposing x2{x}_{2}from the left side, we obtain 0=∑π∈S2∑σ∈S3p2(xπ(1),yσ(1),yσ(2))yσ(3)xπ(3)+p2′(xπ(1),yσ(1),yσ(2))xπ(3)yσ(3)0=\sum _{\pi \in {{\mathbb{S}}}_{2}}\sum _{\sigma \in {{\mathbb{S}}}_{3}}{p}_{2}\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{p}_{2}^{^{\prime} }\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}and p2,p2′:ℒ3→R{p}_{2},{p}_{2}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{3}\to R. Again by using the theory of functional identities and exposing everything from the right side, we obtain 0=∑σ∈S2p2(xπ(1),yσ(1),yσ(2))\hspace{-12.55em}0=\sum _{\sigma \in {{\mathbb{S}}}_{2}}{p}_{2}\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)})and 0=∑σ∈S2p2′(xπ(1),yσ(1),yσ(2)).\hspace{-12.55em}0=\sum _{\sigma \in {{\mathbb{S}}}_{2}}{p}_{2}^{^{\prime} }\left({x}_{\pi \left(1)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}).Therefore, 0=2p2(x,x,x)=2p2′(x,x,x)0=2{p}_{2}\left(x,x,x)=2{p}_{2}^{^{\prime} }\left(x,x,x)for all x∈ℒx\in {\mathcal{ {\mathcal L} }}and p2,p2′:ℒ3→R{p}_{2},{p}_{2}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{3}\to R. Equation (18) can now be rewritten as follows: (20)2xF(x2)−2xF(x)x=n1′(x,x)x+n2′(x,x)x+n3′(x,x)x+λn′(x,x,x)2xF\left({x}^{2})-2xF\left(x)x={n}_{1}^{^{\prime} }\left(x,x)x+{n}_{2}^{^{\prime} }\left(x,x)x+{n}_{3}^{^{\prime} }\left(x,x)x+{\lambda }_{n}^{^{\prime} }\left(x,x,x)for all x∈ℒx\in {\mathcal{ {\mathcal L} }}, ni′:ℒ2→R{n}_{i}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{2}\to R, i=1,2,3i=1,2,3and λn′:ℒ3→C(ℒ){\lambda }_{n}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{3}\to C\left({\mathcal{ {\mathcal L} }}). Now using the theory of functional identities and exposing x1{x}_{1}in (19) from the left side, 0=∑π∈S2∑σ∈S3(p1(xπ(2),yσ(1),yσ(2))yσ(3)xπ(3)−F(yσ(1)yσ(2))xπ(2)yσ(3)xπ(3)+yσ(1)F(yσ(2))xπ(2)yσ(3)xπ(3)+p1′(xπ(2),yσ(1),yσ(2))xπ(3)yσ(3)−yσ(1)F([xπ(2)xπ(3),yσ(2)])yσ(3)+F(yσ(1)yσ(2))xπ(2)xπ(3)yσ(3)−yσ(1)F(yσ(2))xπ(2)xπ(3)yσ(3)+yσ(1)q1(xπ(3),yσ(2),yσ(3))xπ(2)+yσ(1)q2(xπ(2),yσ(2),yσ(3))xπ(3)+yσ(1)q3(xπ(2),xπ(3),yσ(3))yσ(2)+yσ(1)q4(xπ(2),xπ(3),yσ(2))yσ(3)+yσ(1)λq(xπ(2),xπ(3),yσ(2),yσ(3))+yσ(1)xπ(2)F(yσ(2)yσ(3))xπ(3)−yσ(1)xπ(2)F(yσ(2))yσ(3)xπ(3)−F([xπ(2),yσ(1)yσ(2)])yσ(3)xπ(3)\begin{array}{rcl}0& =& \displaystyle \sum _{\pi \in {{\mathbb{S}}}_{2}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}({p}_{1}\left({x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}-F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}+{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(2)}{y}_{\sigma \left(3)}{x}_{\pi \left(3)}\\ & & +{p}_{1}^{^{\prime} }\left({x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}+F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & -{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(2)}{x}_{\pi \left(3)}{y}_{\sigma \left(3)}+{y}_{\sigma \left(1)}{q}_{1}\left({x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)}){x}_{\pi \left(2)}+{y}_{\sigma \left(1)}{q}_{2}\left({x}_{\pi \left(2)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)}){x}_{\pi \left(3)}\\ & & +{y}_{\sigma \left(1)}{q}_{3}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(3)}){y}_{\sigma \left(2)}+{y}_{\sigma \left(1)}{q}_{4}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}+{y}_{\sigma \left(1)}{\lambda }_{q}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)})\\ & & +{y}_{\sigma \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)}){x}_{\pi \left(3)}-{y}_{\sigma \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}{x}_{\pi \left(3)}-F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}\end{array}+yσ(1)F([xπ(2),yσ(2)])yσ(3)xπ(3)−yσ(1)F([xπ(2),yσ(2)yσ(3)])xπ(3)−F([yσ(1)yσ(2),xπ(2)])xπ(3)yσ(3)+yσ(1)yσ(2)F(xπ(2))xπ(3)yσ(3)−yσ(1)yσ(2)F(xπ(2)xπ(3))yσ(3))\begin{array}{rcl}& & +{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}{x}_{\pi \left(3)}-{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}]){x}_{\pi \left(3)}-F\left([{y}_{\sigma \left(1)}{y}_{\sigma \left(2)},{x}_{\pi \left(2)}]){x}_{\pi \left(3)}{y}_{\sigma \left(3)}\\ & & +{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}{y}_{\sigma \left(3)}-{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)}){y}_{\sigma \left(3)})\end{array}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}and pi,pi′,qi,qi′:ℒ3→R{p}_{i},{p}_{i}^{^{\prime} },{q}_{i},{q}_{i}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{3}\to R, i=1,2,3,4i=1,2,3,4and λp,λp′,λq,λq′:ℒ4→C(ℒ){\lambda }_{p},{\lambda }_{p^{\prime} },{\lambda }_{q},{\lambda }_{q^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{4}\to C\left({\mathcal{ {\mathcal L} }}). The aforementioned equation can now be rewritten as (everything exposing from the right side) follows: 0=∑π∈S2∑σ∈S3(yσ(1)q1(xπ(3),yσ(2),yσ(3)))xπ(2)+∑π∈S2∑σ∈S3(yσ(1)q2(xπ(2),yσ(2),yσ(3))+p1(xπ(2),yσ(1),yσ(2))yσ(3)−F(yσ(1)yσ(2))xπ(2)yσ(3)+yσ(1)F(yσ(2))xπ(2)yσ(3)−F([xπ(2),yσ(1)yσ(2)])yσ(3)−yσ(1)xπ(2)F(yσ(2))yσ(3)+yσ(1)xπ(2)F(yσ(2)yσ(3))+yσ(1)F([xπ(2),yσ(2)])yσ(3)−yσ(1)F([xπ(2),yσ(2)yσ(3)]))xπ(3)+∑π∈S2∑σ∈S3(λq(xπ(2),xπ(3),yσ(2),yσ(3)))yσ(1)+∑π∈S2∑σ∈S3(yσ(1)q3(xπ(2),xπ(3),yσ(3)))yσ(2)+∑π∈S2∑σ∈S3(p1′(xπ(2),yσ(1),yσ(2))xπ(3)−F([yσ(1)yσ(2),xπ(2)])xπ(3)−yσ(1)F([xπ(2)xπ(3),yσ(2)])+F(yσ(1)yσ(2))xπ(2)xπ(3)−yσ(1)F(yσ(2))xπ(2)xπ(3)+yσ(1)q4(xπ(2),xπ(3),yσ(2))+yσ(1)yσ(2)F(xπ(2))xπ(3)−yσ(1)yσ(2)F(xπ(2)xπ(3)))yσ(3)\begin{array}{rcl}0& =& \displaystyle \sum _{\pi \in {{\mathbb{S}}}_{2}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}({y}_{\sigma \left(1)}{q}_{1}\left({x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)})){x}_{\pi \left(2)}+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{2}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}({y}_{\sigma \left(1)}{q}_{2}\left({x}_{\pi \left(2)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)})+{p}_{1}\left({x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){y}_{\sigma \left(3)}\\ & & -F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(2)}{y}_{\sigma \left(3)}+{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(2)}{y}_{\sigma \left(3)}-F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}-{y}_{\sigma \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}){y}_{\sigma \left(3)}\\ & & +{y}_{\sigma \left(1)}{x}_{\pi \left(2)}F({y}_{\sigma \left(2)}{y}_{\sigma \left(3)})+{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}]){y}_{\sigma \left(3)}-{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)},{y}_{\sigma \left(2)}{y}_{\sigma \left(3)}])){x}_{\pi \left(3)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{2}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left({\lambda }_{q}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)},{y}_{\sigma \left(3)})){y}_{\sigma \left(1)}+\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{2}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}\left({y}_{\sigma \left(1)}{q}_{3}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(3)})){y}_{\sigma \left(2)}\\ & & +\displaystyle \sum _{\pi \in {{\mathbb{S}}}_{2}}\displaystyle \sum _{\sigma \in {{\mathbb{S}}}_{3}}({p}_{1}^{^{\prime} }\left({x}_{\pi \left(2)},{y}_{\sigma \left(1)},{y}_{\sigma \left(2)}){x}_{\pi \left(3)}-F\left([{y}_{\sigma \left(1)}{y}_{\sigma \left(2)},{x}_{\pi \left(2)}]){x}_{\pi \left(3)}-{y}_{\sigma \left(1)}F\left(\left[{x}_{\pi \left(2)}{x}_{\pi \left(3)},{y}_{\sigma \left(2)}])\\ & & +F({y}_{\sigma \left(1)}{y}_{\sigma \left(2)}){x}_{\pi \left(2)}{x}_{\pi \left(3)}-{y}_{\sigma \left(1)}F({y}_{\sigma \left(2)}){x}_{\pi \left(2)}{x}_{\pi \left(3)}+{y}_{\sigma \left(1)}{q}_{4}\left({x}_{\pi \left(2)},{x}_{\pi \left(3)},{y}_{\sigma \left(2)})+{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}){x}_{\pi \left(3)}-{y}_{\sigma \left(1)}{y}_{\sigma \left(2)}F\left({x}_{\pi \left(2)}{x}_{\pi \left(3)})){y}_{\sigma \left(3)}\end{array}for all x¯3,y¯3∈ℒ3{\bar{x}}_{3},{\bar{y}}_{3}\in {{\mathcal{ {\mathcal L} }}}^{3}and pi,pi′,qi,qi′:ℒ3→R{p}_{i},{p}_{i}^{^{\prime} },{q}_{i},{q}_{i}^{^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{3}\to R, i=1,2,3,4i=1,2,3,4and λp,λp′,λq,λq′:ℒ4→C(ℒ){\lambda }_{p},{\lambda }_{p^{\prime} },{\lambda }_{q},{\lambda }_{q^{\prime} }:{{\mathcal{ {\mathcal L} }}}^{4}\to C\left({\mathcal{ {\mathcal L} }}). Now by using the theory of functional identities, we obtain from the aforementioned equation: q1(x,x,x)=q3(x,x,x)=0{q}_{1}\left(x,x,x)={q}_{3}\left(x,x,x)=0for all x∈ℒx\in {\mathcal{ {\mathcal L} }}and q1,q3:ℒ3→R{q}_{1},{q}_{3}:{{\mathcal{ {\mathcal L} }}}^{3}\to R. Equation (17) can now be rewritten as follows: (21)2F(x2)x−2xF(x)x=xm1(x,x)+xm2(x,x)+xm3(x,x)+λm(x,x,x)2F\left({x}^{2})x-2xF\left(x)x=x{m}_{1}\left(x,x)+x{m}_{2}\left(x,x)+x{m}_{3}\left(x,x)+{\lambda }_{m}\left(x,x,x)for all x∈ℒx\in {\mathcal{ {\mathcal L} }}, mi:ℒ2→R{m}_{i}:{{\mathcal{ {\mathcal L} }}}^{2}\to R, i=1,2,3i=1,2,3and λm:ℒ3→C(ℒ){\lambda }_{m}:{{\mathcal{ {\mathcal L} }}}^{3}\to C\left({\mathcal{ {\mathcal L} }}). Since ℒ{\mathcal{ {\mathcal L} }}is 6-free, after a finite number of steps using equations (20) and (21), we arrive at 2∑(F(xy)−xF(y))=xf(y)+yg(x)+λ(x,y)2\sum \left(F\left(xy)-xF(y))=xf(y)+yg\left(x)+\lambda \left(x,y)2∑(F(x)y−F(xy))=h(y)x+k(x)y+μ(x,y)2\sum \left(F\left(x)y-F\left(xy))=h(y)x+k\left(x)y+\mu \left(x,y)for all x,y∈Rx,y\in R, f,g,h,k:ℒ→Rf,g,h,k:{\mathcal{ {\mathcal L} }}\to Rand λ,μ:ℒ2→C(ℒ)\lambda ,\mu :{{\mathcal{ {\mathcal L} }}}^{2}\to C\left({\mathcal{ {\mathcal L} }}). Therefore, we obtain (22)2F(xy)+2F(yx)−2xF(y)−2yF(x)=xf(y)+yg(x)+λ(x,y)2F\left(xy)+2F(yx)-2xF(y)-2yF\left(x)=xf(y)+yg\left(x)+\lambda \left(x,y)and (23)2F(x)y+2F(y)x−2F(xy)−2F(yx)=h(y)x+k(x)y+μ(x,y).2F\left(x)y+2F(y)x-2F\left(xy)-2F(yx)=h(y)x+k\left(x)y+\mu \left(x,y).For all, x,y∈Rx,y\in R, f,g,h,k:ℒ→Rf,g,h,k:{\mathcal{ {\mathcal L} }}\to Rand λ,μ:ℒ2→C(ℒ)\lambda ,\mu :{{\mathcal{ {\mathcal L} }}}^{2}\to C\left({\mathcal{ {\mathcal L} }}). Replacing the roles of denotations xxand yyin (22) and comparing so obtained identities leads to 0=xf(y)+yg(x)−yf(x)−xg(y)+λ(x,y)−λ(y,x)0=xf(y)+yg\left(x)-yf\left(x)-xg(y)+\lambda \left(x,y)-\lambda (y,x), which yields f(x)=g(x)f\left(x)=g\left(x)and λ(x,y)=λ(y,x)\lambda \left(x,y)=\lambda (y,x)for all x,y∈ℒx,y\in {\mathcal{ {\mathcal L} }}, f,g:ℒ→Rf,g:{\mathcal{ {\mathcal L} }}\to Rand λ:ℒ2→C(ℒ)\lambda :{{\mathcal{ {\mathcal L} }}}^{2}\to C\left({\mathcal{ {\mathcal L} }}). Putting xxfor yyin (22) leads to (24)4F(x2)=4xF(x)+2xf(x)+λ(x,x).4F\left({x}^{2})=4xF\left(x)+2xf\left(x)+\lambda \left(x,x).Using the same arguments, it follows from (23) that h(x)=k(x)h\left(x)=k\left(x)and μ(x,y)=μ(y,x)\mu \left(x,y)=\mu (y,x)for all x,y∈ℒx,y\in {\mathcal{ {\mathcal L} }}, h,k:ℒ→Rh,k:{\mathcal{ {\mathcal L} }}\to Rand μ:ℒ2→C(ℒ)\mu :{{\mathcal{ {\mathcal L} }}}^{2}\to C\left({\mathcal{ {\mathcal L} }}). Therefore, 4F(x2)=4F(x)x−2k(x)x−μ(x,x).4F\left({x}^{2})=4F\left(x)x-2k\left(x)x-\mu \left(x,x).Comparing the aforementioned relations gives 0=x(4F(x)+2f(x))+(−4(x)+2k(x))x+λ(x,x)+μ(x,x).0=x\left(4F\left(x)+2f\left(x))+\left(-4\left(x)+2k\left(x))x+\lambda \left(x,x)+\mu \left(x,x).Hence, there exists r∈Rr\in Rand λ:ℒ→C(ℒ)\lambda :{\mathcal{ {\mathcal L} }}\to C\left({\mathcal{ {\mathcal L} }})such that 4F(x)+2f(x)=rx+λ(x).\hspace{-15.25em}4F\left(x)+2f\left(x)=rx+\lambda \left(x).Considering 2f(x)=−4F(x)+rx+λ(x)2f\left(x)=-4F\left(x)+rx+\lambda \left(x)in (24) gives (25)4F(x2)=xrx+xλ(x)+λ(x,x).\hspace{-17.65em}4F\left({x}^{2})=xrx+x\lambda \left(x)+\lambda \left(x,x).Replacing yyfor xxand xxfor x2{x}^{2}in (22) gives 4F(x3)=2x2F(x)+2xF(x2)+x2f(x)+xf(x2)+λ(x2,x).4F\left({x}^{3})=2{x}^{2}F\left(x)+2xF\left({x}^{2})+{x}^{2}f\left(x)+xf\left({x}^{2})+\lambda \left({x}^{2},x).Using (7) in the aforementioned relation leads to 4F(x2)x−4xF(x)x+4xF(x2)=2x2F(x)+2xF(x2)+x2f(x)+xf(x2)+λ(x2,x).4F\left({x}^{2})x-4xF\left(x)x+4xF\left({x}^{2})=2{x}^{2}F\left(x)+2xF\left({x}^{2})+{x}^{2}f\left(x)+xf\left({x}^{2})+\lambda \left({x}^{2},x).Using (24) in the aforementioned relation leads to 4xf(x)x+3xλ(x,x)=2xf(x2)+2λ(x2,x).4xf\left(x)x+3x\lambda \left(x,x)=2xf\left({x}^{2})+2\lambda \left({x}^{2},x).Considering 2f(x)=−4F(x)+rx+λ(x)2f\left(x)=-4F\left(x)+rx+\lambda \left(x)and using (25) in the aforementioned relation gives 0=−8xF(x)x+xrx2+x2rx+3x2λ(x)+4xλ(x,x)−xλ(x2)−2λ(x2,x).0=-8xF\left(x)x+xr{x}^{2}+{x}^{2}rx+3{x}^{2}\lambda \left(x)+4x\lambda \left(x,x)-x\lambda \left({x}^{2})-2\lambda \left({x}^{2},x).The complete linearization of this relation and using the theory of functional identities leads to 0=−8F(x)x+rx2+xrx+3xλ(x)+4λ(x,x)−λ(x2)0=-8F\left(x)x+r{x}^{2}+xrx+3x\lambda \left(x)+4\lambda \left(x,x)-\lambda \left({x}^{2})and 0=−8F(x)+rx+xr+3λ(x).\hspace{-14em}0=-8F\left(x)+rx+xr+3\lambda \left(x).Therefore, (26)8F(x)=rx+xr+3λ(x).8F\left(x)=rx+xr+3\lambda \left(x).By substituting the aforementioned equation in (7), we obtain 0=−3λ(x3)−6xλ(x2)−3x2λ(x).0=-3\lambda \left({x}^{3})-6x\lambda \left({x}^{2})-3{x}^{2}\lambda \left(x).Since ℒ{\mathcal{ {\mathcal L} }}is a 6-free subset of RR, the aforementioned identity implies λ(x3)=0\lambda \left({x}^{3})=0, λ(x2)=0\lambda \left({x}^{2})=0, λ(x)=0\lambda \left(x)=0for all x∈Rx\in R. From equation (26), we obtain (27)8F(x2)=rx2+x2r.8F\left({x}^{2})=r{x}^{2}+{x}^{2}r.Right (left) multiplication of the relation (26) by xxgives, respectively, (28)8F(x)x=rx2+xrx8F\left(x)x=r{x}^{2}+xrxand (29)8xF(x)=xrx+x2r.8xF\left(x)=xrx+{x}^{2}r.The relations (27)–(29) imply that the additive mapping FFsatisfies the relation 8F(x2)=8F(x)x+8xF(x)−2xrx.\hspace{-14.45em}8F\left({x}^{2})=8F\left(x)x+8xF\left(x)-2xrx.The aforementioned equation can now be rewritten as follows: 4F(x2)=4F(x)x+4xF(x)−xrx\hspace{-14.45em}4F\left({x}^{2})=4F\left(x)x+4xF\left(x)-xrxand (30)4F(x2)=4F(x)x+4xF(x)−2xqx,4F\left({x}^{2})=4F\left(x)x+4xF\left(x)-2xqx,where r=2qr=2q. Let us now introduce the mapping D:R→RD:R\to Rby (31)D(x)=4F(x)−qx−xq.D\left(x)=4F\left(x)-qx-xq.Obviously, the mapping DDis additive. It is our aim to prove that DDis a Jordan derivation. Putting x2{x}^{2}for xxin the aforementioned relation, we obtain D(x2)=4F(x2)−qx2−x2q,D\left({x}^{2})=4F\left({x}^{2})-q{x}^{2}-{x}^{2}q,which gives, after considering the relation (30), the relation (32)D(x2)=4F(x)x+4xF(x)−2xqx−qx2−x2q.D\left({x}^{2})=4F\left(x)x+4xF\left(x)-2xqx-q{x}^{2}-{x}^{2}q.Right (left) multiplication of the relation (31) by xxgives, respectively, (33)D(x)x=4F(x)x−qx2−xqxD\left(x)x=4F\left(x)x-q{x}^{2}-xqxand (34)xD(x)=4xF(x)−xqx−x2q.xD\left(x)=4xF\left(x)-xqx-{x}^{2}q.The relations (32)–(34) imply that the additive mapping DDsatisfies the relation D(x2)=D(x)x+xD(x)\hspace{-12.15em}D\left({x}^{2})=D\left(x)x+xD\left(x)for all x∈Rx\in R. In other words, DDis a Jordan derivation on RR. According to Herstein theorem, one can conclude that DDis a derivation, which completes the proof of the theorem.□We are now in the position to prove Theorem 4.Proof of Theorem 4The complete linearization of (7) gives us (9). First, suppose that RRis not a PI ring (satisfying the standard polynomial identity of degree less than 6). According to Theorem 5, then there exists q∈Rq\in Rsuch that 4F(x)=D(x)+xq+qx4F\left(x)=D\left(x)+xq+qxfor all x∈ℒx\in {\mathcal{ {\mathcal L} }}.Assume now that RRis a PI ring. It is well known that in this case RRhas a nonzero center (see [20]). Let ccbe a nonzero central element. Picking any x∈Rx\in Rand setting x1=x2=cx{x}_{1}={x}_{2}=cxand x3=x{x}_{3}=xin (9), we obtain 3F(c2x3)=F(c2x2)x+2cF(cx2)x−c2xF(x)x−2cxF(cx)x+xF(c2x2)+2cxF(cx2).3F\left({c}^{2}{x}^{3})=F\left({c}^{2}{x}^{2})x+2cF\left(c{x}^{2})x-{c}^{2}xF\left(x)x-2cxF\left(cx)x+xF\left({c}^{2}{x}^{2})+2cxF\left(c{x}^{2}).Next, setting x1=x2=c{x}_{1}={x}_{2}=cand x3=x3{x}_{3}={x}^{3}in (9), we obtain 3F(c2x3)=F(c2)x3+4cF(cx3)−cx3F(c)−cF(c)x3−c2F(x2)x+c2xF(x)x−c2xF(x2)+x3F(c2)3F\left({c}^{2}{x}^{3})=F\left({c}^{2}){x}^{3}+4cF\left(c{x}^{3})-c{x}^{3}F\left(c)-cF\left(c){x}^{3}-{c}^{2}F\left({x}^{2})x+{c}^{2}xF\left(x)x-{c}^{2}xF\left({x}^{2})+{x}^{3}F\left({c}^{2})for all x∈Rx\in R. Comparing both identities, we obtain 0=F(c2)x3+4cF(cx3)−cx3F(c)−cF(c)x3−c2F(x2)x+c2xF(x)x−c2xF(x2)+x3F(c2).0=F\left({c}^{2}){x}^{3}+4cF\left(c{x}^{3})-c{x}^{3}F\left(c)-cF\left(c){x}^{3}-{c}^{2}F\left({x}^{2})x+{c}^{2}xF\left(x)x-{c}^{2}xF\left({x}^{2})+{x}^{3}F\left({c}^{2}).Next, setting x1=x2=x{x}_{1}={x}_{2}=xand x3=c2{x}_{3}={c}^{2}in (9), we obtain (35)3F(c2x2)=2c2F(x2)+2F(c2x)x−c2F(x)x−xF(c2)x−c2xF(x)+2xF(c2x).3F\left({c}^{2}{x}^{2})=2{c}^{2}F\left({x}^{2})+2F\left({c}^{2}x)x-{c}^{2}F\left(x)x-xF\left({c}^{2})x-{c}^{2}xF\left(x)+2xF\left({c}^{2}x).Next, setting x1=x2=x{x}_{1}={x}_{2}=xand x3=c{x}_{3}=cin (9), we obtain (36)3F(cx2)=2cF(x2)+2F(cx)x−cF(x)x−xF(c)x−cxF(x)+2xF(cx).\hspace{-31.5em}3F\left(c{x}^{2})=2cF\left({x}^{2})+2F\left(cx)x-cF\left(x)x-xF\left(c)x-cxF\left(x)+2xF\left(cx).In case x=cx=c, we arrive at F(c3)=2cF(c2)−c2F(c)F\left({c}^{3})=2cF\left({c}^{2})-{c}^{2}F\left(c). Next, setting x1=x2=c{x}_{1}={x}_{2}=cand x3=x{x}_{3}=xin (9), we obtain (37)3F(c2x)=F(c2)x+4cF(cx)−cxF(c)−c2F(x)−cF(c)x+xF(c2).\hspace{-31.5em}3F\left({c}^{2}x)=F\left({c}^{2})x+4cF\left(cx)-cxF\left(c)-{c}^{2}F\left(x)-cF\left(c)x+xF\left({c}^{2}).Setting x1=x2=c{x}_{1}={x}_{2}=cand x3=x2{x}_{3}={x}^{2}in (9), we obtain 3F(c2x2)=F(c2)x2+4cF(cx2)−cx2F(c)−c2F(x2)−cF(c)x2+x2F(c2).\hspace{-31.5em}3F\left({c}^{2}{x}^{2})=F\left({c}^{2}){x}^{2}+4cF\left(c{x}^{2})-c{x}^{2}F\left(c)-{c}^{2}F\left({x}^{2})-cF\left(c){x}^{2}+{x}^{2}F\left({c}^{2}).From the aforementioned equation and (36), we obtain 9F(c2x2)=3F(c2)x2+8c2F(x2)+8cF(cx)x−4c2F(x)x−4cxF(c)x−4c2xF(x)+8cxF(cx)−3cx2F(c)−3c2F(x2)−3cF(c)x2+3x2F(c2).\begin{array}{rcl}9F\left({c}^{2}{x}^{2})& =& 3F\left({c}^{2}){x}^{2}+8{c}^{2}F\left({x}^{2})+8cF\left(cx)x-4{c}^{2}F\left(x)x-4cxF\left(c)x-4{c}^{2}xF\left(x)+8cxF\left(cx)\\ & & -3c{x}^{2}F\left(c)-3{c}^{2}F\left({x}^{2})-3cF\left(c){x}^{2}+3{x}^{2}F\left({c}^{2}).\end{array}Comparing aforementioned equation with (35) and using (37), we obtain (38)0=c2F(x2)−c2F(x)x−c2xF(x)−F(c2)x2−x2F(c2)+cF(c)x2+xF(c2)x+cx2F(c).0={c}^{2}F\left({x}^{2})-{c}^{2}F\left(x)x-{c}^{2}xF\left(x)-F\left({c}^{2}){x}^{2}-{x}^{2}F\left({c}^{2})+cF\left(c){x}^{2}+xF\left({c}^{2})x+c{x}^{2}F\left(c).Complete linearization of the the aforementioned equation and setting x1=c{x}_{1}=cand x2=x{x}_{2}=x, we obtain 0=2c2F(cx)+c2F(c)x−2c3F(x)+c2xF(c)−cxF(c2)−cF(c2)x.0=2{c}^{2}F\left(cx)+{c}^{2}F\left(c)x-2{c}^{3}F\left(x)+{c}^{2}xF\left(c)-cxF\left({c}^{2})-cF\left({c}^{2})x.Setting c=c2c={c}^{2}in (38) we obtain 0=c4F(x2)−c4F(x)x−c4xF(x)−F(c4)x2−x2F(c4)+c2F(c2)x2+xF(c4)x+c2x2F(c2).\hspace{-39.1em}0={c}^{4}F\left({x}^{2})-{c}^{4}F\left(x)x-{c}^{4}xF\left(x)-F\left({c}^{4}){x}^{2}-{x}^{2}F\left({c}^{4})+{c}^{2}F\left({c}^{2}){x}^{2}+xF\left({c}^{4})x+{c}^{2}{x}^{2}F\left({c}^{2}).On other hand, multiplying equation (38) with c2{c}^{2}, we obtain 0=c4F(x2)−c4F(x)x−c4xF(x)−c2F(c2)x2−c2x2F(c2)+c3F(c)x2+c2xF(c2)x+c3x2F(c).0={c}^{4}F\left({x}^{2})-{c}^{4}F\left(x)x-{c}^{4}xF\left(x)-{c}^{2}F\left({c}^{2}){x}^{2}-{c}^{2}{x}^{2}F\left({c}^{2})+{c}^{3}F\left(c){x}^{2}+{c}^{2}xF\left({c}^{2})x+{c}^{3}{x}^{2}F\left(c).Comparing the aforementioned two equations and using F(c4)=3c2F(c2)−2c3F(c)F\left({c}^{4})=3{c}^{2}F\left({c}^{2})-2{c}^{3}F\left(c), we obtain (39)0=−F(c2)x2−x2F(c2)+cF(c)x2+cx2F(c)+2xF(c2)x−2cxF(c)x.0=-F\left({c}^{2}){x}^{2}-{x}^{2}F\left({c}^{2})+cF\left(c){x}^{2}+c{x}^{2}F\left(c)+2xF\left({c}^{2})x-2cxF\left(c)x.On the other hand, adding the same two equations together, we obtain 0=2c2F(x2)−2c2F(x)x−2c2xF(x)−3F(c2)x2−3x2F(c2)+3cF(c)x2+3cx2F(c)+4xF(c2)x−2cxF(c)x.0=2{c}^{2}F\left({x}^{2})-2{c}^{2}F\left(x)x-2{c}^{2}xF\left(x)-3F\left({c}^{2}){x}^{2}-3{x}^{2}F\left({c}^{2})+3cF\left(c){x}^{2}+3c{x}^{2}F\left(c)+4xF\left({c}^{2})x-2cxF\left(c)x.The aforementioned equation can be rewritten as follows: 0=2c2F(x2)−2c2F(x)x−2c2xF(x)−3F(c2)x2−3x2F(c2)+3cF(c)x2+3cx2F(c)+6xF(c2)x−2xF(c2)x−6cxF(c)x+4cxF(c)x.\begin{array}{rcl}0& =& 2{c}^{2}F\left({x}^{2})-2{c}^{2}F\left(x)x-2{c}^{2}xF\left(x)-3F\left({c}^{2}){x}^{2}-3{x}^{2}F\left({c}^{2})+3cF\left(c){x}^{2}+3c{x}^{2}F\left(c)+6xF\left({c}^{2})x\\ & & -2xF\left({c}^{2})x-6cxF\left(c)x+4cxF\left(c)x.\end{array}Using the aforementioned equation and (39), we obtain 0=2c2F(x2)−2c2F(x)x−2c2xF(x)+4cxF(c)x−2xF(c2)x.0=2{c}^{2}F\left({x}^{2})-2{c}^{2}F\left(x)x-2{c}^{2}xF\left(x)+4cxF\left(c)x-2xF\left({c}^{2})x.The aforementioned equation can now be rewritten as follows: c2F(x2)=c2F(x)x+c2xF(x)+xF(c2)x−2cxF(c)x\hspace{-25.15em}{c}^{2}F\left({x}^{2})={c}^{2}F\left(x)x+{c}^{2}xF\left(x)+xF\left({c}^{2})x-2cxF\left(c)xand 4c2F(x2)=4c2F(x)x+4c2xF(x)−2x(2F(c2)−4cF(c))x.\hspace{-25.1em}4{c}^{2}F\left({x}^{2})=4{c}^{2}F\left(x)x+4{c}^{2}xF\left(x)-2x\left(2F\left({c}^{2})-4cF\left(c))x.Setting 2F(c2)−4cF(c)=q2F\left({c}^{2})-4cF\left(c)=q, we arrive at (40)4c2F(x2)=4c2F(x)x+4c2xF(x)−2xqx.4{c}^{2}F\left({x}^{2})=4{c}^{2}F\left(x)x+4{c}^{2}xF\left(x)-2xqx.If q=0q=0than F(x)F\left(x)is derivation, on the other hand, if q≠0q\ne 0, we can conclude as follows. Let us now introduce the mapping D:R→RD:R\to Rby (41)D(x)=4F(x)−qx−xq.D\left(x)=4F\left(x)-qx-xq.Obviously, the mapping DDis additive. It is our aim to prove that DDis a Jordan derivation. Putting x2{x}^{2}for xxin the aforementioned relation, we obtain c2D(x2)=4c2F(x2)−c2qx2−c2x2q,{c}^{2}D\left({x}^{2})=4{c}^{2}F\left({x}^{2})-{c}^{2}q{x}^{2}-{c}^{2}{x}^{2}q,which gives, after considering the relation (40), the relation (42)c2D(x2)=4c2F(x)x+4c2xF(x)−2c2xqx−c2qx2−c2x2q.{c}^{2}D\left({x}^{2})=4{c}^{2}F\left(x)x+4{c}^{2}xF\left(x)-2{c}^{2}xqx-{c}^{2}q{x}^{2}-{c}^{2}{x}^{2}q.Right (left) multiplication of the relation (41) by xxgives, respectively, (43)D(x)x=4F(x)x−qx2−xqxD\left(x)x=4F\left(x)x-q{x}^{2}-xqxand (44)xD(x)=4xF(x)−xqx−x2q.xD\left(x)=4xF\left(x)-xqx-{x}^{2}q.The relations (42)–(44) imply that the additive mapping DDsatisfies the relation c2D(x2)=c2D(x)x+c2xD(x){c}^{2}D\left({x}^{2})={c}^{2}D\left(x)x+{c}^{2}xD\left(x)for all x∈Rx\in R, whence it follows D(x2)=D(x)x+xD(x)D\left({x}^{2})=D\left(x)x+xD\left(x)for all x∈Rx\in R. In other words, DDis a Jordan derivation on RR. According to Herstein theorem, one can conclude that DDis a derivation, which completes the proof of the theorem.□

### Journal

Open Mathematicsde Gruyter

Published: Jan 1, 2022

Keywords: prime ring; derivation; Jordan derivation; two-sided centralizer; functional equation; 16R60; 16W25; 39B05

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