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 ≥ 1 a fixed integer such that [ F ( x ) , y ] n = [ x , F ( y ) ] n , for all x , y ∈ L . Then there exists λ ∈ C such that F ( x ) = λ x , for any x ∈ R , unless R is an order in a 4-dimensional central simple algebra.