Composition of Generalized Derivations as a Lie Derivation

Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, F and G non-zero generalized derivations of R. If the composition (FG) acts as a Lie derivation on R, then (FG) is a derivation of R and one of the following holds: 1. there exist α ∈ C and a ∈ U such that F (x) = [a, x] and G(x) = αx, for all x ∈ R; 2. G is an usual derivation of R and there exists α ∈ C such that F (x) = αx, for all x ∈ R; 3. there exist α, β ∈ C and a derivation h of R such that F (x) = αx+h(x), G(x) = βx, for all x ∈ R, and αβ+h(β) = 0. Moreover in this case h is not an inner derivation of R; 4. there exist a, c ∈ U such that F (x) = ax, G(x) = cx, for all x ∈ R, with ac = 0; 5. there exist b, q ∈ U such that F (x) = xb, G(x) = xq, for all x ∈ R, with qb = 0; 6. there exist c, q ∈ U , η, γ ∈ C such that F (x) = η(xq − cx)+ γx, G(x) = cx+ xq, for all x ∈ R, with γc − ηc′2 = −γq − ηq′2.