Power central valued b-generalized skew derivations on Lie ideals

Let R be a prime ring of characteristic different from 2, \(Q_r\) $Q_r$ be its right Martindale quotient ring and C be its extended centroid. Suppose that F is a non-zero b-generalized skew derivation of R, L a non-central Lie ideal of R and \(n\ge 1\) $n\ge 1$ a fixed positive integer such that \(\biggl (F(x)x\biggr )^n \in Z(R)\) $(F(x)x{)}^n\in Z(R)$ , for all \(x\in L\) $x\in L$ . If either \(char(R)=0\) $char(R)=0$ or \(0\ne char(R)=p \ge n+1\) $0\ne char(R)=p\ge n+1$ , then \(R\subseteq M_2(K)\) $R\subseteq M_2(K)$ , the \(2\times 2\) $2\times 2$ matrix ring over a field K.