On Lie Ideals and Automorphisms in Prime Rings

Let $R$ be a prime ring of characteristic different from $2$ with center $Z$ and extended centroid $C$, and let $L$ be a Lie ideal of $R$. Consider two nontrivial automorphisms $\alpha$ and $\beta$ of $R$ for which there exist integers $m,n\ge 1$ such that $\alpha(u)^n+\beta(u)^m=0$ for all $u\in L$. It is shown that, under these assumptions, either $L$ is central or $R\subseteq M_2(C)$ (where $M_2(C)$ is the ring of $2 \times 2$ matrices over $C$), $L$ is commutative, and $u^{2} \in Z$ for all $u \in L$. In particular, if $L = [R,R]$, then $R$ is commutative.