An engel condition with generalized derivations on lie ideals

Let R be a prime ring, with extended centroid C, g a non-zero generalized derivation of R, L a non-central Lie ideal of R, k a positive fixed integer. If [g(u),u]_k=0, for all u in L, then either there exists a central element c such that g(x)=cx, or R satisfies the standard identity of degree 4. Moreover in the latter case either char(R)=2 or g(x)=ax+xb , where a-b is a central element. We also prove a more generalized version by replacing L with the set [I,I], where I is a right ideal of R.