Generalized derivations as Jordan homomorphisms on lie ideals and right ideals

Let R be a prime ring, L a non-central Lie ideal of R and g a non-zero generalized derivation of R. If g acts as a Jordan homomorphism on L, then either g(x) = x for all x ∈ R, or char(R) = 2, R satisfies the standard identity s ₄(x ₁, x ₂, x ₃, x ₄), L is commutative and u ² ∈ Z(R), for any u ∈ L. We also examine some consequences of this result related to generalized derivations which act as Jordan homomorphisms on the set I, I], where I is a non-zero right ideal of R.