Let m, n be two nonzero fixed positive integers, R a 2-torsion free prime ring with the right Martindale quotient ring Q, L a non-central Lie ideal of R, and δ a derivation of R. Suppose that α is an automorphism of R, D a skew derivation of R with the associated automorphism α, and F a generalized skew derivation of R with the associated skew derivation D. If $$ F(x^{m+n})=F(x^m)x^{n}+x^m\delta(x^{n}) $$ is a polynomial identity for L, then either R satisfies the standard polynomial identity s4(x1, x2, x3, x4) of degree 4, or F is a generalized derivation of R and δ = D. Furthermore, in the latter case one of the following statements holds: (1) D = δ = 0 and there exists a ∈ Q such that F(x) = ax for all x ∈ R; (2) α is the identical mapping of R.