Product of Generalized Derivations with Commuting Values on a Lie Ideal

Let R be a non-commutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, L a non-central Lie ideal of R, F and G two nonzero generalized derivations of R. If \([F(u)G(u),u]=0\) for all \(u\in L\), then one of the following holds: 1. There exist \(u,v \in U\) such that \(uv\in C\) and \(F(x)=xu\), \(G(x)=vx\), for all \(x\in R\); 2. \(R\subseteq M_2(C)\).