Commutators and generalized derivations acting on Lie ideals in prime rings

Let R be a prime ring of char \((R)\ne 2, 3\) and L a noncentral Lie ideal of R. Let U be the Utumi quotient ring of R and \(C=Z(U)\) be the extended centroid of R. Suppose that F, G, H are three generalized derivations of R such that

$$[F(u),u]G(u)+u[H(u),u]=0$$

for all \(u\in L\). Then either R satisfies standard polynomial \(s_4(x_1,x_2,x_3,x_4)\) or one of the following holds:

1. 1.

   There exist \(\alpha , \beta \in C\) such that \(F(x)= \alpha x\) and \(H(x)= \beta x\) for all \( x\in R\);

2. 2.

   There exists \(\beta \in C\) such that \(G(x)=0\), \(H(x)=\beta x\) for all \( x\in R\);

3. 3.

   There exist \(a,b\in U\) and \(0\ne \mu \in C\) such that \(F(x)=xa\), \(G(x)=\mu x\), \(H(x)=bx\) for all \( x\in R\) with \(\mu a+b\in C\).