On a quadratic differential identity with b-generalized derivations.

Let R be a prime ring of characteristic different from 2, with right Martindale quotient ring \(Q_r\) and extended centroid C, L a non-central Lie ideal of R, F a non-zero b-generalized derivation of R and \(\delta \) a non-zero derivation of R, such that \([F(u),\delta (u)]=0\), for all \(u \in L\). Then F is a derivation of R and there exists \(\lambda \in C\) such that \(F=\lambda \delta \), unless when R satisfies the standard identity \(s_4(x_1,\ldots ,x_4)\).