Power values of quadratic polynomials with generalized skew derivations on Lie ideals

Let R be a prime ring of characteristic different from 2, Z(R) its center, Q its right Martindale quotient ring, C its extended centroid, F a generalized skew derivation of R, L a non-central Lie ideal of R and $$n\ge 1$$ a fixed integer such that $$[F(x),y]^n=[x,F(y)]^n$$ , for all $$x,y \in L$$ . Then there exists $$\lambda \in C$$ such that $$F(x)=\lambda x$$ , for any $$x\in R$$ , unless R is an order in a 4-dimensional central simple algebra.