Annihilators on co-commutators with generalized derivations on Lie ideals

Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, L a non-central Lie ideal of R, H and G non-zero generalized derivations of R. Suppose that there exists a non-zero element a of R such that a(H(u)u-uG(u))=0, for all u in L, then one of the following holds: either there exist b,c in U such that H(x)=bx+xc, G(x)=cx with ab=0; or R satisfies the standard identity of degree 4 and there exist b, c, q in U such that H(x)=bx+xc, G(x)=cx+xq, with a(b-q)=0.