Annihilators of power values of generalized skew derivations on Lie ideals in prime rings

Let R be a prime ring of characteristic different from 2, \(n\ge 1\) $n\ge 1$ a fixed integer, C the extended centroid of R, F a generalized skew derivation of R and L a Lie ideal of R. If there exists \(0 \ne a \in R\) $0\ne a\in R$ such that \(a(F(xy)-yx)^{n}=0\) $a{(F(xy)-yx)}^n=0$ for all \(x,y\in L\) $x,y\in L$ , then L is central, unless R satisfies the standard polynomial identity \(s_4(x_1, \ldots , x_4)\) $s_4(x_1,\dots ,x_4)$ .