Annihilators of Skew Derivations with Engel Conditions on Lie Ideals

Let R be a prime ring, L a noncentral Lie ideal of R, and a ∈ R. Set x, y]₁ = x, y] = xy ? yx for x, y ∈ R and inductively x, y]_k = x, y]_k?1, y] for k > 1. Suppose that δ is a nonzero σ-derivation of R such that aδ(x), x]_k = 0 for all x ∈ L, where σ is an automorphism of R and k is a fixed positive integer. Then a = 0 except when char R = 2 and R ? M₂(F), the 2 × 2 matrix ring over a field F.