Commuting Iterates of Generalized Derivations on Lie Ideals

Let R be a prime ring of characteristic different from 2, \(Q_r\) its right Martindale quotient ring, C its extended centroid, F and G two nonzero generalized derivations of R, L a non-central Lie ideal of R and \(n \ge 1\) a fixed integer. If \(\begin{aligned} \Bigl \{F^2(x)x-xG^2(x)\Bigr \}^n=0 \end{aligned}\) for all \(x \in L\) , then either \(R \subseteq M_2(K)\) , the ring \(2 \times 2\) matrices over a field K, or one ot the following holds: