Let R be a prime ring of characteristic different from 2, Q_r its right Martindale quotient ring and C its extended centroid. Suppose that F is a generalized skew derivation of R and L is a non-central Lie ideal of R. If [[F(u),u],F(u)] = 0 for all u ∈ L, then either there exists A ∈ C such that F(x) = Ax, for all x ∈ R or R satisfies S4(x_1,..., x4), the standard identity of degree 4, and there exist a ∈ Q_r and λ ∈ C such that F(x) = ax + xa + Ax, for all x ∈ R.
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