b-generalized skew derivations acting on Lie ideals in prime rings

Let R be any noncommutative prime ring of char(R) ≠ 2, 3, L a noncentral Lie ideal of R and F, G two nonzero b-generalized skew derivations of R. Suppose that \([F(u),\,u]G(u) = 0\) $$[F(u),u]G(u)=0$$ for all u ∈ L. Then at least one of the following conclusions holds: (1)

F(x) = λx for all x ∈ R and for some λ ∈ C, where C is the extended centroid of R