Generalized Derivations with Vanishing Power Values on Lie Ideals

We consider R a prime ring of char R not equal 2 with Utumi quotient ring U and extended centroid C. Let L be a noncentral Lie ideal of R and H a nonzero generalized derivation of R such that (u(1)(n) [H (u), u] u (n)(2)) (n)(3) = 0 for all u is an element of L, where n(1) >= 0, n(2) >= 0, n(3) >= 1 are fixed integers. Then one of the following statements holds: (i) H (x) = a x for all x is an element of R and for some a is an element of C unless R satisfies S 4, the standard identity in four variables; (ii) R satisfies S-4 and H (x) = lambda x - (px + xp) for all x is an element of R and for some lambda is an element of C, p is an element of U