Co-commutators with Generalized Derivations on Lie Ideals in Prime Rings

Let R be a prime ring of characteristic different from 2, La noncentral Lie ideal of R, H and G two nonzero generalized derivations of R. Suppose u(s) (H(u)u-uG(u))u(t) = 0 for all u is an element of L, where s, t >= 0 are fixed integers. Then either (i) there exists p is an element of U such that H(x) = xp for all x is an element of R and G(x) = px for all x is an element of R unless R satisfies S-4 the standard identity in four variables; or (ii) R satisfies S4 and there exist p,q E U such that H(x) = px + xq for all x is an element of R and G(x) = qx + xp for all x is an element of R.