An Engel condition with skew derivations

Let R be a prime ring and set [x, y]₁ = [x, y] = xy ? yx for ${x,yin R}$ and inductively [x, y] _k = [[x, y] _k-1, y] for k > 1. We apply the theory of generalized polynomial identities with automorphisms and skew derivations to obtain the following result: If δ is a nonzero σ-derivation of R and L is a noncommutative Lie ideal of R so that [δ(x), x] _k  = 0 for all ${x in L}$ , where k is a fixed positive integer, then charR = 2 and ${Rsubseteq M_{2}(F)}$ for some field F. This result generalizes the case of derivations by Lanski and also the case of automorphisms by Mayne.