Two generalized derivations on Lie ideals in prime rings

Let $R$ be a prime ring of characteristic not equal to $2$, $U$ be the Utumi quotient ring of $R$ and $C$ be the extended centroid of $R$. Let $G$ and $F$ be two generalized derivations on $R$ and $L$ be a non-central Lie ideal of $R$. If $F\Big(G(u)\Big)u = G(u^{2})$ for all $u \in L$, then one of the following holds: (1) $G=0$. (2) There exist $p,q \in U$ such that $G(x)=p x$, $F(x)=qx$ for all $x \in R$ with $qp=p$. (3) $R$ satisfies $s_4$.