Generalized derivations with annihilating and centralizing Engel conditions on Lie ideals

Let R be a non-commutative prime ring, with center Z(R), extended centroid C and let F be a non-zero generalized derivation of R. Denote by L a non-central Lie ideal of R. If there exists a non-zero element a ∈ R such that a[F(x), x]_k ∈ Z(R) for all x ∈ L, where k is a fixed integer, then one of the followings holds: (1) either there exists λ ∈ C such that F(x) = λx for all x ∈ R; (2) or R satisfies s_4, the standard identity in 4 variables, and char(R) = 2; (3) or R satisfies s_4 and there exist q ∈ U, γ ∈ C such that F(x) = qx + xq + γ x.