Throughout, R will represent an associative ring with center Z(R). As usual the commutator \(xy-yx\) will be denoted by \(\left[ x,y\right] \). We shall use the commutator identity \(\left[ xy,z\right] =\left[ x,z\right] y+x \left[ y,z\right] \). Given an integer \(n\ge 2\), a ring R is said to be n-torsion free, if for \(x\in R,\)\(nx=0\) implies \(x=0.\) Recall that a ring R is prime if for \(a,b\in R,\)\(aRb=(0)\) implies that either \( a=0\) or \(b=0\), and is semiprime in case \(aRa=(0)\) implies \(a=0\). We denote by \(\mathrm{char}(R)\) the characteristic of a ring R. We denote by \(Q_{mr},\)\(Q_{r},Q_{s}\) and C the maximal right ring of quotients, the right ring of quotients, the symmetric Martindale ring of quotients and the extended centroid of a semiprime ring R, respectively. For the explanation of \(Q_{mr},\)\(Q_{r},Q_{s}\) and C, we refer the reader to [3]. An additive mapping \(T{:}\,R\rightarrow R \) is called a left centralizer in case \(T(xy)=T(x)y\) holds for all pairs \(x,y\in R\) and is called a left Jordan centralizer in case \(T(x^{2})=T(x)x\) holds for all \(x\in R\). In case R has the identity element \(T{:}\,R\rightarrow R\) is a left centralizer if T is of the form \(T(x)=ax\) for all \(x\in R\), where \(a\in R\) is a fixed element. For a semiprime ring R all left centralizers are of the form \(T(x)=qx\) for all \(x\in R\), where q is a fixed element from \(Q_{r}\) (see Chapter 2 in [3]). The definition of right centralizer and right Jordan centralizer should be self-explanatory. We call \(T{:}\,R\rightarrow R\) a two-sided centralizer in case T is both a left and a right centralizer. In case \( T{:}\,R\rightarrow R\) is a two-sided centralizer, where R is a semiprime ring with extended centroid C, then there exists element \(\lambda \in C\) such that \(T(x)=\lambda x\) for all \(x\in R\) (see Theorem 2.3.2 in [3]). Zalar [15] has proved that any left (right) Jordan centralizer on a 2-torsion free semiprime ring is a left (right) centralizer (Zalar theorem). For results concerning centralizers (also called multipliers) on rings and algebras we refer to [1, 3, 4, 7, 9, 12, 13] where further references can be found.
Let R be an arbitrary ring and let \(m\ge 0,\)\(n\ge 0\) be some fixed integers with \(m+n\ne 0\). An additive mapping \(T{:}\,R\rightarrow R\) is called an (m, n)-Jordan centralizer in case
$$\begin{aligned} (m+n)T(x^{2})=mT(x)x+nxT(x) \end{aligned}$$
(1)
holds for all \(x\in R\). The concept of (m, n)-Jordan centralizer, which has been introduced by Vukman [14], covers the concept of left Jordan centralizer as well as the concept of right Jordan centralizer. Namely, putting in the relation above \(m=1,\)\(n=0\) one obtains left Jordan centralizer, in case \(m=0,\)\(n=1\) the relation (1) reduces to right Jordan centralizer. In case \(m=n=1\) we obtain the relation
$$\begin{aligned} 2T(x^{2})=T(x)x+xT(x),x\in R. \end{aligned}$$
(2)
Vukman [13] has proved that in case an additive mapping \( T{:}\,R\rightarrow R\), where R is a 2-torsion free semiprime ring, satisfies the relation (2) for any \(x\in R\), then T is a two-sided centralizer. For results concerning (m, n)-Jordan centralizers we refer to [7, 10, 14]. In [14] one can find the following conjecture.
Conjecture 1
[14, Conjecture 2] Let \( m\ge 1,n\ge 1\) be some fixed integers, let R be a semiprime ring with suitable torsion restrictions, and let \(T{:}\,R\rightarrow R\) be an (m, n)-Jordan centralizer. In this case T is a two-sided centralizer.
Kosi-Ulbl and Vukman [8] have recently proved the result below, which proves the conjecture above.
Theorem 2
[8] Let \(m\ge 1,\,n\ge 1\) be some fixed integers, let R be an \(mn(m+n)\)-torsion free semiprime ring, and let \(T{:}\,R\rightarrow R\) be an (m, n)-Jordan centralizer. In this case T is a two-sided centralizer.
Our aim in this paper is to prove the following result.
Theorem 3
Let \(m\ge 1,n\ge 1\) be some fixed integers and let R be a prime ring with \(\mathrm{char}(R)=0\) or \((m+n)^2<\mathrm{char}(R)\). Suppose there exists an additive mapping \(T{:}\,R\rightarrow R\) satisfying the relation
$$\begin{aligned} (m+n)^{2}T(x^{4})=m^{2}T(x)x^{3}+mnxT(x)x^{2}+mnx^{2}T(x)x+n^{2}x^{3}T(x) \end{aligned}$$
(3)
for all \(x\in R.\) In this case T is a two-sided centralizer.
As the main tool in this paper, we use the theory of functional identities (Beidar–Brešar–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 [5] for introductory account on functional identities, where Brešar presents this new theory, the theory of (generalized) functional identities, and its applications, to a wider audience and to [6] for full treatment of this theory.
For the proof of Theorem 3 we need Theorem below, which is of independent interest. Let R be a ring and let
$$\begin{aligned} p(x_1,x_2,x_3,x_4)=\sum _{\pi \in S_3}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \end{aligned}$$
be a fixed multilinear polynomial in noncommutative indeterminates \(x_1\), \(x_2\), \(x_3\) and \(x_4\). Further, let L be a subset of R closed under p i.e., \(p(\overline{x}_4)\in L\) for all \(x_1,x_2,x_3,x_4\in L\), where \(\overline{x}_4=(x_1,x_2,x_3,x_4)\). We shall consider a mapping \(T{:}\,L\rightarrow R\) satisfying
$$\begin{aligned} (m+n)^2T(p(\overline{x}_{4}))= & {} \sum _{\pi \in S_3}(m^2T(x_{\pi (1)})x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&+\,mnx_{\pi (1)}T(x_{\pi (2)})x_{\pi (3)}x_{\pi (4)} +mnx_{\pi (1)}x_{\pi (2)}T( x_{\pi (3)})x_{\pi (4)}\nonumber \\&+\,n^2x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T( x_{\pi (4)})), \end{aligned}$$
(4)
for all \(x_1,x_2,x_3,x_4 \in L\). Of course, every two-sided centralizer satisfies (4). Our goal is to show that under certain assumptions these are in fact the only mappings with this property. In the first step of the proof we derive a functional identity from (4). Let us mention that the idea of considering the expression \([p(\overline{x}_{4}),p(\overline{y}_{4})]\) in its proof is taken from [2].
Theorem 4
Let L be a 8-free Lie subring of R closed under p. If \(T{:}\,L\rightarrow R\) is an additive mapping satisfying (4), then there exist \(p\in C(L)\) and \(\lambda {:}\,L\rightarrow C(L)\) such that \(m^2n(2m+n)T(x)=px+\lambda (x)\) for all \(x\in L\).
FormalPara
Proof
Note that for any \(a\in R\) and \(\overline{x}_{4}\in L^{4}\) we have
$$\begin{aligned} \left[ p(\overline{x}_{4}), a\right]= & {} p\left( [x_1,a], x_{2}, x_{3},x_{3}\right) +p\left( x_1,[x_2,a], x_{3}, x_{4}\right) \\&+\,p\left( x_1,x_2,[x_3,a],x_4\right) + p\left( x_1,x_2,x_3, [x_4,a]\right) . \end{aligned}$$
Thus
$$\begin{aligned} (m+n)^2\left[ p(\overline{x}_{4}), a\right]= & {} (m+n)^2p([x_1,a], x_{2}, x_{3},x_{3})+(m+n)^2p(x_1,[x_2,a], x_{3}, x_{4})\\&+\,(m+n)^2p(x_1,x_2,[x_3,a],x_4)+ (m+n)^2p(x_1,x_2,x_3, [x_4,a]). \end{aligned}$$
Using (4) it follows that
$$\begin{aligned}&(m+n)^2T[p(\overline{x}_{4}), a]=\sum _{\pi \in S_{4}} (m^2T[x_{\pi (1)},a]x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\nonumber \\&\quad +\, mn[x_{\pi (1)},a]T(x_{\pi (2)})x_{\pi (3)}x_{\pi (4)}+\,mn[x_{\pi (1)},a]x_{\pi (2)}T(x_{\pi (3)})x_{\pi (4)}\nonumber \\&\quad +\, n^2[x_{\pi (1)},a]x_{\pi (2)}x_{\pi (3)}T(x_{\pi (4)})+\,m^2T(x_{\pi (1)})[x_{\pi (2)},a]x_{\pi (3)}x_{\pi (4)}\nonumber \\&\quad +\, mnx_{\pi (1)}T[x_{\pi (2)},a]x_{\pi (3)}x_{\pi (4)}+\,mnx_{\pi (1)}[x_{\pi (2)},a]T(x_{\pi (3)})x_{\pi (4)}\nonumber \\&\quad +\, n^2x_{\pi (1)}[x_{\pi (2)},a]x_{\pi (3)}T(x_{\pi (4)}) +\,m^2T(x_{\pi (1)})x_{\pi (2)}[ x_{\pi (3)},a]x_{\pi (4)}\nonumber \\&\quad +\, mnx_{\pi (1)}T(x_{\pi (2)})[ x_{\pi (3)},a]x_{\pi (4)} +\,mnx_{\pi (1)}x_{\pi (2)}T[ x_{\pi (3)},a]x_{\pi (4)}\nonumber \\&\quad +\, n^2x_{\pi (1)}x_{\pi (2)}[x_{\pi (3)},a]T(x_{\pi (4)})+\,m^2T(x_{\pi (1)})x_{\pi (2)}x_{\pi (3)}[x_{\pi (4)},a]\nonumber \\&\quad +\, mnx_{\pi (1)}T(x_{\pi (2)})x_{\pi (3)}[x_{\pi (4)},a] +\,mnx_{\pi (1)}x_{\pi (2)}T(x_{\pi (3)})[x_{\pi (4)},a]\nonumber \\&\quad +\, n^2x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T[x_{\pi (4)},a]). \end{aligned}$$
(5)
Thus
$$\begin{aligned}&(m+n)^2T[p(\overline{x}_{4}), a] =\sum _{\pi \in S_{4}}(m^2T[x_{\pi (1)},a]x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\nonumber \\&\quad +\, m^2T(x_{\pi (1)})[x_{\pi (2)}x_{\pi (3)}x_{\pi (4)},a] +\,mn[x_{\pi (1)},a]T(x_{\pi (2)})x_{\pi (3)}x_{\pi (4)}\nonumber \\&\quad +\,mnx_{\pi (1)}T[x_{\pi (2)},a]x_{\pi (3)}x_{\pi (4)}+\,mnx_{\pi (1)}T(x_{\pi (2)})[ x_{\pi (3)}x_{\pi (4)},a]\nonumber \\&\quad +\, mn[x_{\pi (1)}x_{\pi (2)},a]T(x_{\pi (3)})x_{\pi (4)}+\,mnx_{\pi (1)}x_{\pi (2)}T[ x_{\pi (3)},a]x_{\pi (4)}\nonumber \\&\quad +\,mnx_{\pi (1)}x_{\pi (2)}T(x_{\pi (3)})[x_{\pi (4)},a]+\,n^2[x_{\pi (1)}x_{\pi (2)}x_{\pi (3)},a]T(x_{\pi (4)})\nonumber \\&\quad +\,n^2x_{\pi (1)}x_{\pi (2)} x_{\pi (3)}T[x_{\pi (4)},a]). \end{aligned}$$
(6)
In particular
$$\begin{aligned}&(m+n)^2T[p(\overline{x}_{4}), p(\overline{y}_{4})] =\sum _{\pi \in S_{4}} (m^2T[x_{\pi (1)},p(\overline{y}_{4})]x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\nonumber \\&\quad +\, m^2T(x_{\pi (1)})[x_{\pi (2)}x_{\pi (3)}x_{\pi (4)},p(\overline{y}_{4})]\nonumber \\&\quad +\,mn[x_{\pi (1)},p(\overline{y}_{4})]T(x_{\pi (2)})x_{\pi (3)}x_{\pi (4)} +mnx_{\pi (1)}T[x_{\pi (2)},p(\overline{y}_{4})]x_{\pi (3)}x_{\pi (4)}\nonumber \\&\quad +\,mnx_{\pi (1)}T(x_{\pi (2)})[ x_{\pi (3)}x_{\pi (4)},p(\overline{y}_{4})] + mn[x_{\pi (1)}x_{\pi (2)},p(\overline{y}_{4})]T(x_{\pi (3)})x_{\pi (4)} \nonumber \\&\quad +\,mnx_{\pi (1)}x_{\pi (2)}T[ x_{\pi (3)},p(\overline{y}_{4})]x_{\pi (4)} +mnx_{\pi (1)}x_{\pi (2)}T(x_{\pi (3)})[x_{\pi (4)},p(\overline{y}_{4})] \nonumber \\&\quad +\,n^2[x_{\pi (1)}x_{\pi (2)}x_{\pi (3)},p(\overline{y}_{4})]T(x_{\pi (4)}) +n^2x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T[x_{\pi (4)},p(\overline{y}_{4})]). \end{aligned}$$
(7)
for all \(\overline{x}_{4},\overline{y}_{4}\in L^4\). For \(i=1,2,3,4\) we also have [by (7)]
$$\begin{aligned}&(m+n)^2T[x_{\pi (i)}, p(\overline{y}_{4})]= \sum _{\sigma \in S_{4}} (m^2T[x_{\pi (i)},y_{\sigma (1)}]y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)} \nonumber \\&\quad +\,m^2T(y_{\sigma (1)})[x_{\pi (i)},y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}] + mn[x_{\pi (i)},y_{\sigma (1)}]T(y_{\sigma (2)})y_{\sigma (3)}y_{\sigma (4)} \nonumber \\&\quad +\,mny_{\sigma (1)}T[x_{\pi (i)},y_{\sigma (2)}]y_{\sigma (3)}y_{\sigma (4)}+mny_{\sigma (1)}T(y_{\sigma (2)})[x_{\pi (i)}, y_{\sigma (3)}y_{\sigma (4)}] \nonumber \\&\quad +\,mn[x_{\pi (i)},y_{\sigma (1)}y_{\sigma (2)}]T(y_{\sigma (3)})y_{\sigma (4)} + mny_{\sigma (1)}y_{\sigma (2)}T[x_{\pi (i)}, y_{\sigma (3)}]y_{\sigma (4)}\nonumber \\&\quad +\,mny_{\sigma (1)}y_{\sigma (2)}T(y_{\sigma (3)})[x_{\pi (i)},y_{\sigma (4)}] + n^2[x_{\pi (i)},y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}]T(y_{\sigma (4)}) \nonumber \\&\quad +\,n^2y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T[x_{\pi (i)},y_{\sigma (4)}]). \end{aligned}$$
(8)
Equation (7) can now be written as
$$\begin{aligned}&(m+n)^4T[p(\overline{x}_{4}), p(\overline{y}_{4})] = (m+n)^2 (\sum _{\pi \in S_{4}} ( m^2T[x_{\pi (1)},p(\overline{y}_{4})]x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\,m^2T(x_{\pi (1)})[x_{\pi (2)}x_{\pi (3)}x_{\pi (4)},p(\overline{y}_{4})] +mn[x_{\pi (1)},p(\overline{y}_{4})]T(x_{\pi (2)})x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\,mnx_{\pi (1)}T[x_{\pi (2)},p(\overline{y}_{4})]x_{\pi (3)}x_{\pi (4)}+mnx_{\pi (1)}T(x_{\pi (2)})[ x_{\pi (3)}x_{\pi (4)},p(\overline{y}_{4})]\nonumber \\&\quad +\,mn[x_{\pi (1)}x_{\pi (2)},p(\overline{y}_{4})]T(x_{\pi (3)})x_{\pi (4)} +mnx_{\pi (1)}x_{\pi (2)}T[ x_{\pi (3)},p(\overline{y}_{4})]x_{\pi (4)} \nonumber \\&\quad +\,mnx_{\pi (1)}x_{\pi (2)}T(x_{\pi (3)})[x_{\pi (4)},p(\overline{y}_{4})] + n^2[x_{\pi (1)}x_{\pi (2)}x_{\pi (3)},p(\overline{y}_{4})]T(x_{\pi (4)}) \nonumber \\&\quad +\, n^2x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T[x_{\pi (4)},p(\overline{y}_{4})])). \end{aligned}$$
(9)
for all \(\overline{x}_{4},\overline{y}_{4}\in L^4\).
Therefore last equation can be written as
$$\begin{aligned}&(m+n)^4T[p(\overline{x}_{4}), p(\overline{y}_{4})]\nonumber \\&\quad =\sum _{\pi \in S_{4}}\sum _{\sigma \in S_{4}} ( ( m^4T[x_{\pi (1)},y_{\sigma (1)}]y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)} + m^4T(y_{\sigma (1)})[x_{\pi (1)},y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}] \nonumber \\&\quad +\, m^3n[x_{\pi (1)},y_{\sigma (1)}]T(y_{\sigma (2)})y_{\sigma (3)}y_{\sigma (4)} +m^3ny_{\sigma (1)}T[x_{\pi (1)},y_{\sigma (2)}]y_{\sigma (3)}y_{\sigma (4)}\nonumber \\&\quad +\,m^3ny_{\sigma (1)}T(y_{\sigma (2)})[x_{\pi (1)}, y_{\sigma (3)}y_{\sigma (4)}] + m^3n[x_{\pi (1)},y_{\sigma (1)}y_{\sigma (2)}]T(y_{\sigma (3)})y_{\sigma (4)} \nonumber \\&\quad +\,m^3ny_{\sigma (1)}y_{\sigma (2)}T[x_{\pi (1)}, y_{\sigma (3)}]y_{\sigma (4)} +m^3ny_{\sigma (1)}y_{\sigma (2)}T(y_{\sigma (3)})[x_{\pi (1)},y_{\sigma (4)}] \nonumber \\&\quad +\, m^2n^2[x_{\pi (1)},y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}]T(y_{\sigma (4)}) +m^2n^2y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T[x_{\pi (1)},y_{\sigma (4)}] ) x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\, (m+n)^2m^2T(x_{\pi (1)})[x_{\pi (2)}x_{\pi (3)}x_{\pi (4)},y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}] \nonumber \\&\quad +\, (m+n)^2mn[x_{\pi (1)},y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}]T(x_{\pi (2)})x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\,x_{\pi (1)} ( m^3nT[x_{\pi (2)},y_{\sigma (1)}]y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)} + m^3nT(y_{\sigma (1)})[x_{\pi (2)},y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}] \nonumber \\&\quad +\, m^2n^2[x_{\pi (2)},y_{\sigma (1)}]T(y_{\sigma (2)})y_{\sigma (3)}y_{\sigma (4)} +m^2n^2y_{\sigma (1)}T[x_{\pi (2)},y_{\sigma (2)}]y_{\sigma (3)}y_{\sigma (4)}\nonumber \\&\quad +\,m^2n^2y_{\sigma (1)}T(y_{\sigma (2)})[x_{\pi (2)}, y_{\sigma (3)}y_{\sigma (4)}] + m^2n^2[x_{\pi (2)},y_{\sigma (1)}y_{\sigma (2)}]T(y_{\sigma (3)})y_{\sigma (4)} \nonumber \\&\quad +\,m^2n^2y_{\sigma (1)}y_{\sigma (2)}T[x_{\pi (2)}, y_{\sigma (3)}]y_{\sigma (4)} +m^2n^2y_{\sigma (1)}y_{\sigma (2)}T(y_{\sigma (3)})[x_{\pi (2)},y_{\sigma (4)}] \nonumber \\&\quad +\, mn^3[x_{\pi (2)},y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}]T(y_{\sigma (4)}) +n^2y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T[x_{\pi (2)},y_{\sigma (4)}] ) x_{\pi (3)}x_{\pi (4)}\nonumber \\&\quad +\,(m+n)^2mnx_{\pi (1)}T(x_{\pi (2)})[ x_{\pi (3)}x_{\pi (4)},y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}] \nonumber \\&\quad +\, (m+n)^2mn[x_{\pi (1)}x_{\pi (2)},y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}]T(x_{\pi (3)})x_{\pi (4)} \nonumber \\&\quad +\,x_{\pi (1)}x_{\pi (2)} ( m^3nT[x_{\pi (3)},y_{\sigma (1)}]y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)} + m^3nT(y_{\sigma (1)})[x_{\pi (3)},y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}] \nonumber \\&\quad +\, m^2n^2[x_{\pi (3)},y_{\sigma (1)}]T(y_{\sigma (2)})y_{\sigma (3)}y_{\sigma (4)} +m^2n^2y_{\sigma (1)}T[x_{\pi (3)},y_{\sigma (2)}]y_{\sigma (3)}y_{\sigma (4)}\nonumber \\&\quad +\,m^2n^2y_{\sigma (1)}T(y_{\sigma (2)})[x_{\pi (3)}, y_{\sigma (3)}y_{\sigma (4)}] + m^2n^2[x_{\pi (3)},y_{\sigma (1)}y_{\sigma (2)}]T(y_{\sigma (3)})y_{\sigma (4)} \nonumber \\&\quad +\,m^2n^2y_{\sigma (1)}y_{\sigma (2)}T[x_{\pi (3)}, y_{\sigma (3)}]y_{\sigma (4)} +m^2n^2y_{\sigma (1)}y_{\sigma (2)}T(y_{\sigma (3)})[x_{\pi (3)},y_{\sigma (4)}] \nonumber \\&\quad +\, mn^3[x_{\pi (3)},y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}]T(y_{\sigma (4)}) +mn^3y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T[x_{\pi (3)},y_{\sigma (4)}] ) x_{\pi (4)} \nonumber \\&\quad +\,(m+n)^2mnx_{\pi (1)}x_{\pi (2)}T(x_{\pi (3)})[x_{\pi (4)},y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}] \nonumber \\&\quad +\, (m+n)^2n^2[x_{\pi (1)}x_{\pi (2)}x_{\pi (3)},y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}]T(x_{\pi (4)}) \nonumber \\&\quad +\,x_{\pi (1)}x_{\pi (2)}x_{\pi (3)} ( m^2n^2T[x_{\pi (4)},y_{\sigma (1)}]y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)} + m^2n^2T(y_{\sigma (1)})[x_{\pi (4)},y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}] \nonumber \\&\quad +\, mn^3[x_{\pi (4)},y_{\sigma (1)}]T(y_{\sigma (2)})y_{\sigma (3)}y_{\sigma (4)} +mn^3y_{\sigma (1)}T[x_{\pi (4)},y_{\sigma (2)}]y_{\sigma (3)}y_{\sigma (4)}\nonumber \\&\quad +\,mn^3y_{\sigma (1)}T(y_{\sigma (2)})[x_{\pi (4)}, y_{\sigma (3)}y_{\sigma (4)}] + mn^3[x_{\pi (4)},y_{\sigma (1)}y_{\sigma (2)}]T(y_{\sigma (3)})y_{\sigma (4)} \nonumber \\&\quad +\,mn^3y_{\sigma (1)}y_{\sigma (2)}T[x_{\pi (4)}, y_{\sigma (3)}]y_{\sigma (4)} +mn^3y_{\sigma (1)}y_{\sigma (2)}T(y_{\sigma (3)})[x_{\pi (4)},y_{\sigma (4)}] \nonumber \\&\quad +\, n^4[x_{\pi (4)},y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}]T(y_{\sigma (4)}) +n^4y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T[x_{\pi (4)},y_{\sigma (4)}] ) ). \end{aligned}$$
(10)
On the other hand, using \([p(\overline{x}_{4}),p(\overline{y}_{4})]=-[p(\overline{y}_{4}), p(\overline{x}_{4})],\) we get from the above identity
$$\begin{aligned}&(m+n)^4T[p(\overline{x}_{4}),p(\overline{y}_{4}) ]\nonumber \\&=\sum _{\pi \in S_{4}}\sum _{\pi \in S_{4}} (( m^4T[x_{\pi (1)},y_{\sigma (1)}]x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} + m^4T(x_{\pi (1)})[x_{\pi (2)}x_{\pi (3)}x_{\pi (4)},y_{\sigma (1)}] \nonumber \\&\quad +\, m^3n[x_{\pi (1)},y_{\sigma (1)}]T(x_{\pi (2)})x_{\pi (3)}x_{\pi (4)} +m^3nx_{\pi (1)}T[x_{\pi (2)},y_{\sigma (1)}]x_{\pi (3)}x_{\pi (4)}\nonumber \\&\quad +\,m^3nx_{\pi (1)}T(x_{\pi (2)})[x_{\pi (3)}x_{\pi (4)},y_{\sigma (1)} ] + m^3n[x_{\pi (1)}x_{\pi (2)},y_{\sigma (1)}]T(x_{\pi (3)})x_{\pi (4)} \nonumber \\&\quad +\,m^3nx_{\pi (1)}x_{\pi (2)}T[x_{\pi (3)},y_{\sigma (1)} ]x_{\pi (4)} +m^3nx_{\pi (1)}x_{\pi (2)}T(x_{\pi (3)})[x_{\pi (4)},y_{\sigma (1)}] \nonumber \\&\quad +\, m^2n^2[x_{\pi (1)}x_{\pi (2)}x_{\pi (3)},y_{\sigma (1)}]T(x_{\pi (4)}) +m^2n^2x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T[x_{\pi (4)},y_{\sigma (1)}] ) y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)} \nonumber \\&\quad +\, (m+n)^2m^2T(y_{\sigma (1)})[x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)},y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}] \nonumber \\&\quad +\, (m+n)^2mn[x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)},y_{\sigma (1)}]T(y_{\sigma (2)})y_{\sigma (3)}y_{\sigma (4)} \nonumber \\&\quad +\,y_{\sigma (1)} ( m^3nT[x_{\pi (1)},y_{\sigma (2)}]x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} + m^3nT(x_{\pi (1)})[x_{\pi (2)}x_{\pi (3)}x_{\pi (4)},y_{\sigma (2)}] \nonumber \\&\quad +\, m^2n^2[x_{\pi (1)},y_{\sigma (2)}]T(x_{\pi (2)})x_{\pi (3)}x_{\pi (4)} +m^2n^2x_{\pi (1)}T[x_{\pi (2)},y_{\sigma (2)}]x_{\pi (3)}x_{\pi (4)}\nonumber \\&\quad +\,m^2n^2x_{\pi (1)}T(x_{\pi (2)})[x_{\pi (3)}x_{\pi (4)},y_{\sigma (2)}] + m^2n^2[x_{\pi (1)}x_{\pi (2)},y_{\sigma (2)}]T(x_{\pi (3)})x_{\pi (4)} \nonumber \\&\quad +\,m^2n^2x_{\pi (1)}x_{\pi (2)}T[x_{\pi (3)},y_{\sigma (2)} ]x_{\pi (4)} +m^2n^2x_{\pi (1)}x_{\pi (2)}T(x_{\pi (3)})[x_{\pi (4)},y_{\sigma (2)}] \nonumber \\&\quad +\, mn^3[x_{\pi (1)}x_{\pi (2)}x_{\pi (3)},y_{\sigma (2)}]T(x_{\pi (4)}) +mn^3x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T[x_{\pi (4)},y_{\sigma (2)}] ) y_{\sigma (3)}y_{\sigma (4)}\nonumber \\&\quad +\,(m+n)^2mny_{\sigma (1)}T(y_{\sigma (2)})[x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}, y_{\sigma (3)}y_{\sigma (4)}] \nonumber \\&\quad +\, (m+n)^2mn[x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)},y_{\sigma (1)}y_{\sigma (2)}]T(y_{\sigma (3)})y_{\sigma (4)} \nonumber \\&\quad +\,y_{\sigma (1)}y_{\sigma (2)} ( m^3nT[x_{\pi (1)},y_{\sigma (3)}]x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} + m^3nT(x_{\pi (1)})[x_{\pi (2)}x_{\pi (3)}x_{\pi (4)},y_{\sigma (3)}] \nonumber \\&\quad +\, m^2n^2[x_{\pi (1)},y_{\sigma (3)}]T(x_{\pi (2)})x_{\pi (3)}x_{\pi (4)} +m^2n^2x_{\pi (1)}T[x_{\pi (2)},y_{\sigma (3)}]x_{\pi (3)}x_{\pi (4)}\nonumber \\&\quad +\,m^2n^2x_{\pi (1)}T(x_{\pi (2)})[x_{\pi (3)}x_{\pi (4)},y_{\sigma (3)}] + m^2n^2[x_{\pi (1)}x_{\pi (2)},y_{\sigma (3)}]T(x_{\pi (3)})x_{\pi (4)} \nonumber \\&\quad +\,m^2n^2x_{\pi (1)}x_{\pi (2)}T[x_{\pi (3)},y_{\sigma (3)} ]x_{\pi (4)} +m^2n^2x_{\pi (1)}x_{\pi (2)}T(x_{\pi (3)})[x_{\pi (4)},y_{\sigma (3)}] \nonumber \\&\quad +\, mn^3[x_{\pi (1)}x_{\pi (2)}x_{\pi (3)},y_{\sigma (3)}]T(x_{\pi (4)}) +mn^3x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T[x_{\pi (4)},y_{\sigma (3)}] ) y_{\sigma (4)} \nonumber \\&\quad +\,(m+n)^2mny_{\sigma (1)}y_{\sigma (2)}T(y_{\sigma (3)})[x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)},y_{\sigma (4)}] \nonumber \\&\quad +\, (m+n)^2n^2[x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)},y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}]T(y_{\sigma (4)}) \nonumber \\&\quad +\,y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)} ( m^2n^2T[x_{\pi (1)},y_{\sigma (4)}]x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} + m^2n^2T(x_{\pi (1)})[x_{\pi (2)}x_{\pi (3)}x_{\pi (4)},y_{\sigma (4)}] \nonumber \\&\quad +\, mn^3[x_{\pi (1)},y_{\sigma (4)}]T(x_{\pi (2)})x_{\pi (3)}x_{\pi (4)} +mn^3x_{\pi (1)}T[x_{\pi (2)},y_{\sigma (4)}]x_{\pi (3)}x_{\pi (4)}\nonumber \\&\quad +\,mn^3x_{\pi (1)}T(x_{\pi (2)})[x_{\pi (3)}x_{\pi (4)},y_{\sigma (4)} ] + mn^3[x_{\pi (1)}x_{\pi (2)},y_{\sigma (4)}]T(x_{\pi (3)})x_{\pi (4)} \nonumber \\&\quad +\,mn^3x_{\pi (1)}x_{\pi (2)}T[x_{\pi (3)},y_{\sigma (4)} ]x_{\pi (4)} +mn^3x_{\pi (1)}x_{\pi (2)}T(x_{\pi (3)})[x_{\pi (4)},y_{\sigma (4)}] \nonumber \\&\quad +\, n^4[x_{\pi (1)}x_{\pi (2)}x_{\pi (3)},y_{\sigma (4)}]T(x_{\pi (4)}) +n^4x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T[x_{\pi (4)},y_{\sigma (4)}] ) ). \end{aligned}$$
(11)
for all \(\overline{x}_{4},\overline{y}_{4}\in L^4\).
Comparing so obtained identities, we arrive at
$$\begin{aligned} 0= & {} \sum _{\pi \in S_{4}}\sum _{\sigma \in S_{4}} ( x_{\pi (1)} ( (m+n)^2n^2 x_{\pi (2)}x_{\pi (3)}y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}T(x_{\pi (4)}) \nonumber \\&\quad -\,(2mn^3+m^2n^2) x_{\pi (2)}x_{\pi (3)} x_{\pi (4)}y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T(y_{\sigma (4)}) \nonumber \\&\quad -\,n^4 x_{\pi (2)}x_{\pi (3)}y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)} x_{\pi (4)}T(y_{\sigma (4)}) \nonumber \\&\quad +\,n^4 x_{\pi (2)}x_{\pi (3)} y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T[x_{\pi (4)},y_{\sigma (4)}] \nonumber \\&\quad +\, (m^3n-m^2n^2) x_{\pi (1)}y_{\sigma (1)}T(y_{\sigma (2)})y_{\sigma (3)}y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\,(m^3n-m^2n^2) x_{\pi (1)}y_{\sigma (1)}y_{\sigma (2)}T(y_{\sigma (3)})y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\, m^2n^2x_{\pi (1)}y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T(y_{\sigma (4)})x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\, (m+n)^2mn x_{\pi (1)}y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}T(x_{\pi (2)})x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\,m^3n x_{\pi (1)}T[x_{\pi (2)},y_{\sigma (1)}]y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\, m^3n x_{\pi (1)} T(y_{\sigma (1)})x_{\pi (2)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad -\, m^3n x_{\pi (1)} T(y_{\sigma (1)})y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad -\, m^2n^2 x_{\pi (1)}y_{\sigma (1)}x_{\pi (2)}T(y_{\sigma (2)})y_{\sigma (3)}y_{\sigma (4)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\,m^2n^2x_{\pi (1)}y_{\sigma (1)}T[x_{\pi (2)},y_{\sigma (2)}]y_{\sigma (3)}y_{\sigma (4)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\,m^2n^2x_{\pi (1)}y_{\sigma (1)}T(y_{\sigma (2)})x_{\pi (2)}y_{\sigma (3)}y_{\sigma (4)}x_{\pi (3)}x_{\pi (4)} \\ \end{aligned}$$
$$\begin{aligned}&\quad -\, m^2n^2 x_{\pi (1)}y_{\sigma (1)}y_{\sigma (2)}x_{\pi (2)}T(y_{\sigma (3)})y_{\sigma (4)} x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\,m^2n^2 x_{\pi (1)} y_{\sigma (1)}y_{\sigma (2)}T[x_{\pi (2)}, y_{\sigma (3)}]y_{\sigma (4)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\,m^2n^2 x_{\pi (1)}y_{\sigma (1)}y_{\sigma (2)}T(y_{\sigma (3)})x_{\pi (2)}y_{\sigma (4)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\, mn^3 x_{\pi (1)} x_{\pi (2)}y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T(y_{\sigma (4)})x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad -\, mn^3 x_{\pi (1)}y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)} x_{\pi (2)}T(y_{\sigma (4)})x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\,n^2 x_{\pi (1)} y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T[x_{\pi (2)},y_{\sigma (4)}]x_{\pi (3)}x_{\pi (4)}\nonumber \\&\quad -\,(m+n)^2mn x_{\pi (1)}T(x_{\pi (2)}) y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\, (m+n)^2mn x_{\pi (1)}x_{\pi (2)}y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}T(x_{\pi (3)})x_{\pi (4)} \nonumber \\&\quad +\,m^3n x_{\pi (1)}x_{\pi (2)}T[x_{\pi (3)},y_{\sigma (1)}]y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}x_{\pi (4)} \nonumber \\&\quad +\, m^3n x_{\pi (1)}x_{\pi (2)} T(y_{\sigma (1)})x_{\pi (3)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)} x_{\pi (4)} \nonumber \\&\quad -\, m^3n x_{\pi (1)}x_{\pi (2)} T(y_{\sigma (1)})y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)} x_{\pi (3)} x_{\pi (4)} \nonumber \\&\quad +\, (m^2n^2 -mn^3)x_{\pi (1)}x_{\pi (2)} x_{\pi (3)}y_{\sigma (1)}T(y_{\sigma (2)})y_{\sigma (3)}y_{\sigma (4)}x_{\pi (4)} \nonumber \\&\quad -\,m^2n^2 x_{\pi (1)}x_{\pi (2)}y_{\sigma (1)} x_{\pi (3)}T(y_{\sigma (2)})y_{\sigma (3)}y_{\sigma (4)}x_{\pi (4)} \nonumber \\&\quad +\,m^2n^2x_{\pi (1)}x_{\pi (2)}y_{\sigma (1)}T[x_{\pi (3)},y_{\sigma (2)}]y_{\sigma (3)}y_{\sigma (4)}x_{\pi (4)} \nonumber \\&\quad +\,m^2n^2 x_{\pi (1)}x_{\pi (2)} y_{\sigma (1)}T(y_{\sigma (2)})x_{\pi (3)} y_{\sigma (3)}y_{\sigma (4)}x_{\pi (4)} \nonumber \\&\quad +\, (m^2n^2-mn^3) x_{\pi (1)}x_{\pi (2)} x_{\pi (3)}y_{\sigma (1)}y_{\sigma (2)}T(y_{\sigma (3)})y_{\sigma (4)}x_{\pi (4)} \nonumber \\ \end{aligned}$$
$$\begin{aligned}&\quad -\, m^2n^2 x_{\pi (1)}x_{\pi (2)} y_{\sigma (1)}y_{\sigma (2)}x_{\pi (3)}T(y_{\sigma (3)})y_{\sigma (4)}x_{\pi (4)} \nonumber \\&\quad +\,m^2n^2x_{\pi (1)}x_{\pi (2)}y_{\sigma (1)}y_{\sigma (2)}T[x_{\pi (3)}, y_{\sigma (3)}]y_{\sigma (4)}x_{\pi (4)} \nonumber \\&\quad +\,m^2n^2x_{\pi (1)}x_{\pi (2)}y_{\sigma (1)}y_{\sigma (2)}T(y_{\sigma (3)})x_{\pi (3)}y_{\sigma (4)}x_{\pi (4)} \nonumber \\&\quad +\, mn^3 x_{\pi (1)}x_{\pi (2)} x_{\pi (3)}y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T(y_{\sigma (4)})x_{\pi (4)} \nonumber \\&\quad -\, mn^3 x_{\pi (1)}x_{\pi (2)}y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)} x_{\pi (3)}T(y_{\sigma (4)})x_{\pi (4)} \nonumber \\&\quad -\, m^2n^2 x_{\pi (1)}x_{\pi (2)}x_{\pi (3)} T(y_{\sigma (1)})y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}x_{\pi (4)} \nonumber \\&\quad -\,(m+n)^2mn x_{\pi (1)}x_{\pi (2)}T(x_{\pi (3)})y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}x_{\pi (4)} \nonumber \\&\quad +\,mn^3x_{\pi (1)}x_{\pi (2)}y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T[x_{\pi (3)},y_{\sigma (4)}]x_{\pi (4)} )\nonumber \\&\quad +\, ( m^4T[x_{\pi (1)},y_{\sigma (1)}]y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)} \nonumber \\&\quad +\, m^4T(y_{\sigma (1)})x_{\pi (1)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)} \nonumber \\&\quad +\,(2m^3n+m^2n^2)T(y_{\sigma (1)})y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)} \nonumber \\&\quad -\, (m+n)^2m^2 T(x_{\pi (1)})y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)} ) x_{\pi (4)} \nonumber \\&\quad +\, y_{\sigma (1)} ( (m+n)^2n^2 y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}T(y_{\sigma (4)}) \nonumber \\&\quad -\, (2mn^3+m^2n^2) y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T(x_{\pi (4)}) \nonumber \\&\quad -\, n^4 y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}y_{\sigma (4)}T(x_{\pi (4)}) \nonumber \\&\quad -\,n^4 y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T[x_{\pi (4)},y_{\sigma (4)}] \nonumber \\&\quad -\, m^3n x_{\pi (1)}T(y_{\sigma (2)})y_{\sigma (3)}y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\,m^3n T[x_{\pi (1)},y_{\sigma (2)}]y_{\sigma (3)}y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\,m^3n T(y_{\sigma (2)})x_{\pi (1)} y_{\sigma (3)}y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\,(2m^2n^2+mn^3) T(y_{\sigma (2)})y_{\sigma (3)}y_{\sigma (4)}x_{\pi (1)} x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad -\,m^3n y_{\sigma (2)}x_{\pi (1)}T(y_{\sigma (3)})y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\,m^3ny_{\sigma (2)}T[x_{\pi (1)}, y_{\sigma (3)}]y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\,m^3ny_{\sigma (2)}T(y_{\sigma (3)})x_{\pi (1)}y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\,(2m^2n^2+mn^3) y_{\sigma (2)}T(y_{\sigma (3)})y_{\sigma (4)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\ \end{aligned}$$
$$\begin{aligned}&\quad -\, m^2n^2y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}T(y_{\sigma (4)})x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad -\, (2m^2n^2+m^3n) y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}x_{\pi (1)}T(x_{\pi (2)})x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad -\, (2m^2n^2+m^3n) y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}x_{\pi (1)}x_{\pi (2)}T(x_{\pi (3)})x_{\pi (4)} \nonumber \\&\quad +\, m^2n^2 y_{\sigma (2)}y_{\sigma (3)}T(x_{\pi (1)})y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad -\, mn^3 y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}y_{\sigma (4)}T(x_{\pi (2)})x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad -\,mn^3 y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}T[x_{\pi (2)},y_{\sigma (4)}]x_{\pi (3)}x_{\pi (4)}\nonumber \\&\quad +\,mn^3 y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}T(x_{\pi (2)})y_{\sigma (4)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad -\, mn^3 y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}x_{\pi (2)}y_{\sigma (4)}T(x_{\pi (3)})x_{\pi (4)} \nonumber \\&\quad -\,mn^3 y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}x_{\pi (2)}T[x_{\pi (3)},y_{\sigma (4)} ]x_{\pi (4)} \nonumber \\&\quad +\,mn^3 y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}x_{\pi (2)}T(x_{\pi (3)})y_{\sigma (4)}x_{\pi (4)} ) \nonumber \\&\quad +\, ((2m^3n+m^2n^2) T(x_{\pi (1)})x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)} \nonumber \\&\quad +\,(2m^2n^2+mn^3) x_{\pi (1)}T(x_{\pi (2)}) x_{\pi (3)}x_{\pi (4)}y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)} \nonumber \\&\quad +\,(2m^2n^2+mn^3) x_{\pi (1)}x_{\pi (2)}T(x_{\pi (3)})x_{\pi (4)}y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)} \nonumber \\&\quad +\,m^2n^2 x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T[x_{\pi (4)},y_{\sigma (1)}]y_{\sigma (2)}y_{\sigma (3)} \nonumber \\&\quad +\, m^2n^2 x_{\pi (1)}x_{\pi (2)}x_{\pi (3)} T(y_{\sigma (1)})x_{\pi (4)}y_{\sigma (2)}y_{\sigma (3)} \nonumber \\&\quad -\,( 2m^2n^2+m^3n) x_{\pi (1)}x_{\pi (2)}x_{\pi (3)} x_{\pi (4)}y_{\sigma (1)}T(y_{\sigma (2)})y_{\sigma (3)} \nonumber \\&\quad -\, mn^3 x_{\pi (1)}x_{\pi (2)}x_{\pi (3)} y_{\sigma (1)}x_{\pi (4)}T(y_{\sigma (2)})y_{\sigma (3)} \nonumber \\&\quad +\,mn^3 x_{\pi (1)}x_{\pi (2)}x_{\pi (3)} y_{\sigma (1)}T[x_{\pi (4)},y_{\sigma (2)}]y_{\sigma (3)}\nonumber \\&\quad +\,mn^3 x_{\pi (1)}x_{\pi (2)}x_{\pi (3)} y_{\sigma (1)}T(y_{\sigma (2)})x_{\pi (4)} y_{\sigma (3)} \nonumber \\&\quad -\, (2m^2n^2+m^3n) x_{\pi (1)}x_{\pi (2)}x_{\pi (3)} x_{\pi (4)}y_{\sigma (1)}y_{\sigma (2)}T(y_{\sigma (3)}) \nonumber \\&\quad -\, mn^3 x_{\pi (1)}x_{\pi (2)}x_{\pi (3)} y_{\sigma (1)}y_{\sigma (2)}x_{\pi (4)}T(y_{\sigma (3)}) \nonumber \\&\quad +\,mn^3 x_{\pi (1)}x_{\pi (2)}x_{\pi (3)} y_{\sigma (1)}y_{\sigma (2)}T[x_{\pi (4)}, y_{\sigma (3)}] \nonumber \\&\quad +\,mn^3 x_{\pi (1)}x_{\pi (2)}x_{\pi (3)} y_{\sigma (1)}y_{\sigma (2)}T(y_{\sigma (3)})x_{\pi (4)} \nonumber \\&\quad -\, m^4T[x_{\pi (1)},y_{\sigma (1)}]x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} y_{\sigma (2)}y_{\sigma (3)} \nonumber \\&\quad +\, m^4T(x_{\pi (1)})y_{\sigma (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}y_{\sigma (2)}y_{\sigma (3)} \nonumber \\&\quad -\, m^3n x_{\pi (1)}y_{\sigma (1)} T(x_{\pi (2)})x_{\pi (3)}x_{\pi (4)}y_{\sigma (2)}y_{\sigma (3)} \nonumber \\&\quad -\, (m+n)^2m^2T(y_{\sigma (1)})x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}y_{\sigma (2)}y_{\sigma (3)} \nonumber \\ \end{aligned}$$
$$\begin{aligned}&\quad +\, (m+n)^2mn y_{\sigma (1)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}T(y_{\sigma (2)})y_{\sigma (3)} \nonumber \\&\quad +\, (m^3n-m^2n^2) y_{\sigma (1)}x_{\pi (1)}T(x_{\pi (2)})x_{\pi (3)}x_{\pi (4)}y_{\sigma (2)}y_{\sigma (3)} \nonumber \\&\quad -\,m^3n x_{\pi (1)}T[x_{\pi (2)},y_{\sigma (1)}]x_{\pi (3)}x_{\pi (4)}y_{\sigma (2)}y_{\sigma (3)} \nonumber \\&\quad +\,m^3n x_{\pi (1)}T(x_{\pi (2)})y_{\sigma (1)} x_{\pi (3)}x_{\pi (4)}y_{\sigma (2)}y_{\sigma (3)} \nonumber \\&\quad -\, m^3n x_{\pi (1)}x_{\pi (2)}y_{\sigma (1)}T(x_{\pi (3)})x_{\pi (4)} y_{\sigma (2)}y_{\sigma (3)} \nonumber \\&\quad +\, (m^3n-m^2n^2) y_{\sigma (1)}x_{\pi (1)}x_{\pi (2)}T(x_{\pi (3)})x_{\pi (4)} y_{\sigma (2)}y_{\sigma (3)} \nonumber \\&\quad -\,m^3n x_{\pi (1)}x_{\pi (2)}T[x_{\pi (3)},y_{\sigma (1)} ]x_{\pi (4)}y_{\sigma (2)}y_{\sigma (3)} \nonumber \\&\quad +\,m^3nx_{\pi (1)}x_{\pi (2)}T(x_{\pi (3)})y_{\sigma (1)}x_{\pi (4)}y_{\sigma (2)}y_{\sigma (3)} \nonumber \\&\quad -\, m^2n^2 x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}y_{\sigma (1)}T(x_{\pi (4)}) y_{\sigma (2)}y_{\sigma (3)} \nonumber \\&\quad +\, m^2n^2 y_{\sigma (1)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T(x_{\pi (4)}) y_{\sigma (2)}y_{\sigma (3)} \nonumber \\&\quad -\,m^2n^2x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T[x_{\pi (4)},y_{\sigma (1)}])y_{\sigma (2)}y_{\sigma (3)} \nonumber \\&\quad -\,m^3n y_{\sigma (1)} T[x_{\pi (1)},y_{\sigma (2)}] x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}y_{\sigma (3)} \nonumber \\&\quad -\, m^3n y_{\sigma (1)} T(x_{\pi (1)})x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}y_{\sigma (2)} y_{\sigma (3)} \nonumber \\&\quad +\, m^3n y_{\sigma (1)}T(x_{\pi (1)})y_{\sigma (2)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} y_{\sigma (3)} \nonumber \\&\quad -\, m^2n^2 y_{\sigma (1)}x_{\pi (1)}y_{\sigma (2)}T(x_{\pi (2)})x_{\pi (3)}x_{\pi (4)} y_{\sigma (3)} \nonumber \\&\quad -\,m^2n^2 y_{\sigma (1)}x_{\pi (1)}T[x_{\pi (2)},y_{\sigma (2)}]x_{\pi (3)}x_{\pi (4)}y_{\sigma (3)} \nonumber \\&\quad +\,m^2n^2y_{\sigma (1)}x_{\pi (1)}T(x_{\pi (2)})y_{\sigma (2)}x_{\pi (3)}x_{\pi (4)} y_{\sigma (3)} \nonumber \\&\quad -\, m^2n^2 y_{\sigma (1)}x_{\pi (1)}x_{\pi (2)}y_{\sigma (2)}T(x_{\pi (3)})x_{\pi (4)} y_{\sigma (3)} \nonumber \\&\quad -\,m^2n^2 y_{\sigma (1)}x_{\pi (1)}x_{\pi (2)}T[x_{\pi (3)},y_{\sigma (2)} ]x_{\pi (4)} y_{\sigma (3)} \nonumber \\&\quad +\,m^2n^2y_{\sigma (1)}x_{\pi (1)}x_{\pi (2)}T(x_{\pi (3)})y_{\sigma (2)}x_{\pi (4)} y_{\sigma (3)} \nonumber \\&\quad -\, mn^3y_{\sigma (1)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}y_{\sigma (2)}T(x_{\pi (4)}) y_{\sigma (3)} \nonumber \\&\quad +\, mn^3y_{\sigma (1)}y_{\sigma (2)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T(x_{\pi (4)}) y_{\sigma (3)} \nonumber \\&\quad -\,mn^3y_{\sigma (1)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T[x_{\pi (4)},y_{\sigma (2)}])y_{\sigma (3)}\nonumber \\&\quad -\,(m+n)^2mn y_{\sigma (1)}T(y_{\sigma (2)})x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} y_{\sigma (3)} \nonumber \\&\quad +\, (m+n)^2mn y_{\sigma (1)}y_{\sigma (2)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}T(y_{\sigma (3)}) \nonumber \\&\quad -\,m^3n y_{\sigma (1)}y_{\sigma (2)}T[x_{\pi (1)},y_{\sigma (3)}]x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad -\, m^3n y_{\sigma (1)}y_{\sigma (2)}T(x_{\pi (1)})x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}y_{\sigma (3)} \nonumber \\&\quad +\, m^3n y_{\sigma (1)}y_{\sigma (2)}T(x_{\pi (1)})y_{\sigma (3)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad -\, m^2n^2 y_{\sigma (1)}y_{\sigma (2)}x_{\pi (1)}y_{\sigma (3)}T(x_{\pi (2)})x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\, (m^2n^2 -mn^3) y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}T(x_{\pi (2)})x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad -\,m^2n^2 y_{\sigma (1)}y_{\sigma (2)}x_{\pi (1)}T[x_{\pi (2)},y_{\sigma (3)}]x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\,m^2n^2 y_{\sigma (1)}y_{\sigma (2)}x_{\pi (1)}T(x_{\pi (2)})y_{\sigma (3)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad -\, m^2n^2 y_{\sigma (1)}y_{\sigma (2)}x_{\pi (1)}x_{\pi (2)}y_{\sigma (3)}T(x_{\pi (3)})x_{\pi (4)} \nonumber \\&\quad +\, (m^2n^2-mn^3) y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}x_{\pi (2)}T(x_{\pi (3)})x_{\pi (4)} \nonumber \\&\quad -\,m^2n^2 y_{\sigma (1)}y_{\sigma (2)}x_{\pi (1)}x_{\pi (2)}T[x_{\pi (3)},y_{\sigma (3)} ]x_{\pi (4)} \nonumber \\&\quad +\,m^2n^2 y_{\sigma (1)}y_{\sigma (2)}x_{\pi (1)}x_{\pi (2)}T(x_{\pi (3)})y_{\sigma (3)}x_{\pi (4)} \nonumber \\&\quad -\, mn^3 y_{\sigma (1)}y_{\sigma (2)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}y_{\sigma (3)}T(x_{\pi (4)}) \nonumber \\&\quad +\, mn^3 y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T(x_{\pi (4)}) \nonumber \\&\quad -\,mn^3 y_{\sigma (1)}y_{\sigma (2)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T[x_{\pi (4)},y_{\sigma (3)}]) \nonumber \\&\quad -\,(m+n)^2mn y_{\sigma (1)}y_{\sigma (2)}T(y_{\sigma (3)})x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad -\, m^2n^2 y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T(x_{\pi (1)})x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}) y_{\sigma (4)}). \end{aligned}$$
(12)
for all \(x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4 \in L\). Let \(s{:}\,\mathbb {Z} \rightarrow \mathbb {Z}\) be a mapping defined by \(s(i)=i-3\). For each \(\sigma \in S_4 \) the mapping \(s^{-1}\sigma s{:}\,\{5,6,7,8\} \rightarrow \{5,6,7,8\}\) will be denoted by \(\overline{\sigma }\). Writing \(x_{4+i}\) instead of \(y_i\), \(i=1,2,3,4\), in the above identity we can express this relation as
$$\begin{aligned} \sum _{i=1}^8E_i^i(\overline{x}_8)x_i + \sum _{j=1}^8x_jF_j^j(\overline{x}_8)=0. \end{aligned}$$
For example
$$\begin{aligned} E_4^4(\overline{x}_8)= & {} \sum _{\begin{array}{c} \pi \in S_{4}\\ {\pi }(4)=4 \end{array} } \sum _{\sigma \in S_{4}} (m^4T[x_{\pi (1)},x_{\overline{\sigma }(5)}]x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (2)}x_{\pi (3)} \nonumber \\&+\,m^4T(x_{\overline{\sigma }(5)})x_{\pi (1)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (2)}x_{\pi (3)} \nonumber \\&+\,(2m^3n+m^2n^2)T(x_{\overline{\sigma }(5)})x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)} \nonumber \\&-\,(m+n)^2m^2 T(x_{\pi (1)})x_{\overline{\sigma }(5)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (2)}x_{\pi (3)} ), \end{aligned}$$
(13)
and
$$\begin{aligned}&F_5^5(\overline{x}_8)= \sum _{\pi \in S_{4}} \sum _{\begin{array}{c} \sigma \in S_{4}\\ {\overline{\sigma }}(5)=5 \end{array}} {\Big (} (m+n)^2n^2 x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}T(x_{\overline{\sigma }(8)}) \nonumber \\&\quad -\, (2mn^3+m^2n^2) x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T(x_{\pi (4)}) \nonumber \\&\quad +\,(2m^2n^2+mn^3) T(x_{\overline{\sigma }(6)})x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (1)} x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\,(2m^2n^2+mn^3) x_{\overline{\sigma }(6)}T(x_{\overline{\sigma }(7)})x_{\overline{\sigma }(8)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad -\, (2m^2n^2+m^3n) x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (1)}T(x_{\pi (2)})x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad -\, (2m^2n^2+m^3n) x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (1)}x_{\pi (2)}T(x_{\pi (3)})x_{\pi (4)} \nonumber \\&\quad -\, n^4 x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\overline{\sigma }(8)}T(x_{\pi (4)})\nonumber \\&\quad -\,n^4 x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T[x_{\pi (4)},x_{\overline{\sigma }(8)}] \nonumber \\&\quad -\, m^3n x_{\pi (1)}T(x_{\overline{\sigma }(6)})x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\nonumber \\&\quad +\,m^3n T[x_{\pi (1)},x_{\overline{\sigma }(6)}]x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\,m^3n T(x_{\overline{\sigma }(6)})x_{\pi (1)} x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\nonumber \\&\quad -\,m^3n x_{\overline{\sigma }(6)}x_{\pi (1)}T(x_{\overline{\sigma }(7)})x_{\overline{\sigma }(8)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\,m^3nx_{\overline{\sigma }(6)}T[x_{\pi (1)}, x_{\overline{\sigma }(7)}]x_{\overline{\sigma }(8)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\nonumber \\&\quad +\,m^3nx_{\overline{\sigma }(6)}T(x_{\overline{\sigma }(7)})x_{\pi (1)}x_{\overline{\sigma }(8)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad +\, m^2n^2 x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}T(x_{\pi (1)})x_{\overline{\sigma }(8)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\nonumber \\&\quad -\,mn^3 x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\pi (1)}x_{\overline{\sigma }(8)}T(x_{\pi (2)})x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad -\,mn^3 x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\pi (1)}T[x_{\pi (2)},x_{\overline{\sigma }(8)}]x_{\pi (3)}x_{\pi (4)}\nonumber \\&\quad +\,mn^3 x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\pi (1)}T(x_{\pi (2)})x_{\overline{\sigma }(8)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&\quad -\, mn^3 x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\pi (1)}x_{\pi (2)}x_{\overline{\sigma }(8)}T(x_{\pi (3)})x_{\pi (4)}\nonumber \\&\quad -\,mn^3 x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\pi (1)}x_{\pi (2)}T[x_{\pi (3)},x_{\overline{\sigma }(8)} ]x_{\pi (4)} \nonumber \\&\quad -\, m^2n^2x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\pi (1)}T(x_{\overline{\sigma }(8)})x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\nonumber \\&\quad +\,mn^3 x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\pi (1)}x_{\pi (2)}T(x_{\pi (3)})x_{\overline{\sigma }(8)}x_{\pi (4)} {\Big )} \end{aligned}$$
(14)
for all \(\overline{x}_8 \in L^8\). Since L is 8-free, it follows that the functional identity (13) has only a standard solution. In particular, there exist mappings \(p_{4,j}{:}\,L^6\rightarrow R\), \(j=1,2,3,5,6,7,8\) and \(\lambda _{4}{:}\,L^7\rightarrow C(L)\) such that
$$\begin{aligned} E_4^4(\overline{x}_8)= \sum _{j=1}^8 x_jp_{4,j}^{4,j}(\overline{x}_8)+\lambda _4^4(\overline{x}_8) \end{aligned}$$
for all \(\overline{x}_8\in L^8\). Note that this is also a functional identity which can be rewritten as
$$\begin{aligned}&\sum _{\begin{array}{c} \pi \in S_{4}\\ {\pi }(4)=4 \end{array} } \sum _{\sigma \in S_{4}} (( m^4T[x_{\pi (1)},x_{\overline{\sigma }(5)}]x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (2)} \nonumber \\&\quad +\,m^4T(x_{\overline{\sigma }(5)})x_{\pi (1)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (2)} \nonumber \\&\quad +\,(2m^3n+m^2n^2)T(x_{\overline{\sigma }(5)})x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (1)}x_{\pi (2)} \nonumber \\&\quad -\,(m+n)^2m^2 T(x_{\pi (1)})x_{\overline{\sigma }(5)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (2)} ) x_{\pi (3)})\nonumber \\&\quad -\,\sum _{j=1}^8 x_jp_{4,j}^{4,j}(\overline{x}_8) \in C(L) \end{aligned}$$
(15)
for all \(\overline{x}_8\in L^8\). Thus
$$\begin{aligned} E_1^1(\overline{x}_7)x_1 + E_2^2(\overline{x}_7)x_2 + E_3^3(\overline{x}_7)x_3 - \sum _{j=1}^8 x_jp_{4,j}^{4,j}(\overline{x}_8) \in C(L), \end{aligned}$$
for all \(\overline{x}_7 \in L^7\), where for example
$$\begin{aligned} E_3^3(\overline{x}_7)= & {} \sum _{\begin{array}{c} \pi \in S_{4}\\ {\pi }(4)=4 \end{array} } \sum _{\sigma \in S_{4}} (( m^4T[x_{\pi (1)},x_{\overline{\sigma }(5)}]x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)} \\&+ m^4T(x_{\overline{\sigma }(5)})x_{\pi (1)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)} \\&+\,(2m^3n+m^2n^2)T(x_{\overline{\sigma }(5)})x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (1)} \\&-\,(m+n)^2m^2 T(x_{\pi (1)})x_{\overline{\sigma }(5)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)} ) x_{\pi (2)}) \end{aligned}$$
for all \(\overline{x}_5\in L^5\). Again using that L is 8-free, this identity has only a standard solution. Hence there are mappings \(p_{3,j}{:}\,L^5 \rightarrow R\), \(j=1,2,5,6,7,8\) such that
$$\begin{aligned} E_3^3(\overline{x}_7)-\sum _{j=1}^8 x_jp_{3,j}^{3,j}(\overline{x}_7)\in C(L). \end{aligned}$$
We continue with the same procedure as above. Finally there exist \(p\in R\) and a mapping \(\lambda {:}\,L\rightarrow C(L)\) such that
$$\begin{aligned} m^2n(2m+n)T(x)=xp+\lambda (x) \end{aligned}$$
(16)
for all \(x\in L\). Similarly, using (14) and the same method as above, we can show that there exist \(q\in R\) and a mapping \(\mu {:}\,L\rightarrow C(L)\) such that
$$\begin{aligned} mn^2(2n+m) T(x)=qx+\mu (x) \end{aligned}$$
(17)
for all \(x\in L\). Thus
$$\begin{aligned} m^2n^2(2m+n)(2n+m)T(x)= & {} (2n+m)nxp+(2n+m)n\lambda (x)\\ m^2n^2(2n+m)(2m+n) T(x)= & {} (2m+n)mqx+(2m+n)m\mu (x) \end{aligned}$$
Comparing the above two identities we arrive at
$$\begin{aligned} 0=(2n+m)nxp-(2m+n)mqx + (2n+m)n \lambda (x)-(2m+n)m \mu (x) \end{aligned}$$
for all \(x\in L\). It follows that \(p\in C(L)\), \((2n+m)np=(2m+n)mq\) and \((2n+m)n\lambda (x) =(2m+n)m\mu (x)\). Thereby the proof is completed. \(\square \)
We are now in the position to prove Theorem 5.
Proof
The complete linearization of (3) gives us (4).
First suppose that R is not a PI ring (satisfying the standard polynomial identity of degree less than 12). According to Theorem there exist \(p\in C\) and \(\lambda {:}\,R\rightarrow C\) such that
$$\begin{aligned} m^2n(2m+n)T(x)=xp+\lambda (x) \end{aligned}$$
for all \(x\in R\). Using this with (4) we see that
$$\begin{aligned} (m+n)^2\lambda (p(\overline{x}_4))= & {} \sum _{\pi \in S_4} m^2 \lambda (x_{\pi (1)})x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\nonumber \\&+\sum _{\pi \in S_4} mn x_{\pi (1)}\lambda (x_{\pi (2)})x_{\pi (3)}x_{\pi (4)} \nonumber \\&+\,\sum _{\pi \in S_4} mnx_{\pi (1)}x_{\pi (2)}\lambda (x_{\pi (3)})x_{\pi (4)}\nonumber \\&+\sum _{\pi \in S_4} n^2x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}\lambda (x_{\pi (4)}) \nonumber \\= & {} (m+n)^2\sum _{\pi \in S_3} \lambda (x_{\pi (1)})x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \end{aligned}$$
(18)
for all \(x_1,x_2,x_3,x_4 \in R\). Note that (18) can also be written as
$$\begin{aligned} \sum _{\pi \in S_4} \lambda (x_{\pi (1)})x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \in C \end{aligned}$$
for all \(x_1,x_2,x_3,x_4 \in R\). Now using the theory of functional identities it follows that \(\lambda (R)=0\) and so \(\lambda =0\). Thus T is a two-sided centralizer, as desired.
Assume now that R is a PI ring. It is well-known that in this case R has a nonzero center (see [9]). Let c be a nonzero central element. Pick any \(x\in R\) and set \(x_1=x_2=x_3=cx\) and \(x_4=x\) in complete linearization of (3). We arrive at
$$\begin{aligned} (m+n)^{2}4T(c^3x^4)= & {} m^2c^3T(x)x^3+3m^2c^2T(cx)x^3+mnc^2xT(cx)x^2\\&+\,mn c^3xT(x)x^2+2mnc^2 xT(cx)x^2+mnc^2x^2T(cx)x\\&+\,mnc^3x^2T(x)x+2mnc^2x^2T(cx)x\\&+\,n^2c^3x^3T(x)+3n^2c^2x^3T(cx). \end{aligned}$$
On the other hand, setting \(x_1=x_2=x_3=c\) and \(x_4=x^4\) in (4) we obtain
$$\begin{aligned} (m+n)^{2}4T(c^3x^4)= & {} (m+n)^2c^3T(x^4)+(3m^2+3mn)c^2T(c)x^4\\&+\,(3mn+3n^2)c^2x^4T(c). \end{aligned}$$
Comparing so obtained relations and using (3) we get
$$\begin{aligned} 0= & {} m^2T(cx)x^3+mnxT(cx)x^2+mnx^2T(cx)x \nonumber \\&+\,n^2x^3T(cx)-(m^2+mn)T(c)x^4-(mn+n^2)x^4T(c). \end{aligned}$$
(19)
In the case when \(x=c\) we have
$$\begin{aligned} T(c^2)=cT(c). \end{aligned}$$
(20)
Now setting \(x_1=x_2=c\), \(x_3=c^2\) and \(x_4=x\) in complete linearization of (3) we obtain
$$\begin{aligned} (m+n)^24T(c^4x)= & {} (m+n)^2c^4T(x)+(mn+n^2)c^2xT(c^2)\\&+\,(mn+m^2)c^2T(c^2)x+(2mn+2n^2)c^3xT(c)\\&+\,(2mn+2m^2)c^3T(c)x \end{aligned}$$
On the other hand, setting \(x_1=x_2=x_3=c\) and \(x_4=cx\) in complete linearization of (3) we obtain
$$\begin{aligned} (m+n)^{2}4T(c^4x)= & {} (m+n)^2c^3T(cx)\\&+\,(3m^2+3mn)c^3T(c)x+(3mn+3n^2)c^3xT(c). \end{aligned}$$
Comparing so obtained relations and using (20) we get
$$\begin{aligned} T(cx)=cT(x). \end{aligned}$$
(21)
Using (19) and (21) we arrive at
$$\begin{aligned} 0= & {} m^2cT(x)x^3+mncxT(x)x^2+mncx^2T(x)x \nonumber \\&+\,n^2cx^3T(x)-(m^2+mn)T(c)x^4-(mn+n^2)x^4T(c). \end{aligned}$$
(22)
for all \(x\in R\). Now setting \(x_1=x_2=x_3=c\) and \(x_4=x\) in complete linearization of (22) we obtain
$$\begin{aligned} (m+n)T(cx)=mT(c)x+nxT(c) \end{aligned}$$
(23)
for all \(x\in R\). Using Eq. (23) in (19) we arrive at
$$\begin{aligned} 0= & {} (-2m-n)T(c)x^4+(-2n-m)x^4T(c)+2mxT(c)x^3 \nonumber \\&+\,2nx^3T(c)x+(n+m)x^2T(c)x^2. \end{aligned}$$
(24)
for all \(x\in R\). Now setting \(x_1=x_2=x\) and \(x_3=x_4=c\) in complete linearization of (24) we obtain
$$\begin{aligned} 2(20m+20n)xT(c)x=(20m+20n)T(c)x^2+(20m+20n)x^2T(c) \end{aligned}$$
and therefore
$$\begin{aligned} 2xT(c)x=T(c)x^2+x^2T(c) \end{aligned}$$
(25)
for all \(x \in R\). The last equation can now be rewritten as
$$\begin{aligned}{}[[T(c),x],x]=0 \end{aligned}$$
(26)
Now using second Posner theorem [11], we can conclude that \([T(c), x] = 0\) for all \( x\in R\). From Eq. (23) we now obtain
$$\begin{aligned} T(cx) = T(c)x = xT(c). \end{aligned}$$
(27)
Now putting x instead of xy in Eq. (23) we get
$$\begin{aligned}&(m + n)T(xy)c = (mT(c)x)y + x(nyT(c))\nonumber \\&\quad = (m + n)T(x)yc + (m + n)xT(y)c - (m + n)xT(c)y. \end{aligned}$$
(28)
Multiplying last equation on the left by z we get
$$\begin{aligned}&(m + n)zT(xy)c \nonumber \\&\quad =(m + n)zT(x)yc + (m + n)zxT(y)c- (m + n)zxT(c)y. \end{aligned}$$
(29)
Replacing x with zx in Eq. (28) we get
$$\begin{aligned}&(m + n)T(zxy)c \nonumber \\&\quad =(m + n)T(zx)yc+ (m + n)zxT(y)c -(m + n)zxT(c)y. \end{aligned}$$
(30)
Comparing Eqs. (29) and (30) it follows
$$\begin{aligned} T(zxy) = zT(xy) + T(zx)y -zT(x)y. \end{aligned}$$
(31)
Now replacing x with c in the last equation we obtain
$$\begin{aligned} T(zcy) = zT(cy) + T(zc)y -zT(c)y. \end{aligned}$$
Using (27) in the last equation we get
$$\begin{aligned} T(zcy) = zT(cy) + zT(c)y- zT(c)y \end{aligned}$$
and by (21)
$$\begin{aligned} T(zy)c = zT(y)c. \end{aligned}$$
Now it follows that \(T(zy) = zT(y)\) for all \(y, z \in R\). Similarly we can prove that \(T(zy) = T(z)y\) for all \(y, z\in R\). Multiplying Eq. (28) on the right side by z we get
$$\begin{aligned}&(m + n)T(xy)cz \nonumber \\&\quad =(m + n)T(x)ycz + (m + n)xT(y)cz- (m + n)xT(c)yz. \end{aligned}$$
(32)
Replacing y with yz in Eq. (28) we get
$$\begin{aligned}&(m + n)T(xyz)c \nonumber \\&\quad =(m + n)T(x)yzc+ (m + n)xT(yz)c -(m + n)xT(c)yz. \end{aligned}$$
(33)
Comparing Eqs. (32) and (33) it follows
$$\begin{aligned} T(xyz) = T(xy)z + xT(yz) -xT(y)z. \end{aligned}$$
(34)
Now replacing y with c in the last equation and using (27) and (21) we obtain
$$\begin{aligned} T(xz) = T(x)z \end{aligned}$$
for all \(x,z \in R\). Thus T is a two-sided centralizer. Thereby the proof is completed. \(\square \)
