An equation concerning power values of generalized skew derivations and annihilating conditions on Lie ideals

Let R be a prime ring of characteristic different from 2, Qr its right Martindale quotient ring and C its extended centroid. Suppose that F is a generalized skew derivation of R, L a non-central Lie ideal of R, a. R a non-zero element of R, m, r >= 1 and s, t >= 0 fixed integers. If a (F(u)(r) (F(u)(s)u + uF(u)(s)) F(u)(t))(m) = 0 for all u is an element of L, then either there exists b. Qr such that F(x) = bx, for any x is an element of R, with ab = 0 or R subset of M-2(K), the ring of 2 x 2 matrices over a field K.