Generalized Skew Derivations with Periodic Values on Lie Ideals

Let R be a prime non commutative ring, L a noncentral Lie ideal of R, \(\alpha \) an automorphism of R, \(G: R\longrightarrow R\) a non-zero generalized skew derivation of R, \(m,n\ge 2\) fixed positive integers, such that \(\begin{aligned}\biggl (G(u)^n-G(u)\biggr )^m=0,\,\,\,\text { for all }u\in L.\end{aligned}\) If \(char(R)=0\) or \(char(R)=p>0\) where \(p \not \mid 2^{n}-2\) , then \(R\subseteq M_2(C)\) , the ring of \(2\times 2\) matrices over the extended centroid C.