Let R be a prime ring of characteristic different from 2 with center Z and extended centroid C, and let L be a Lie ideal of R. Consider two nontrivial automorphisms α and β of R for which there exist integers m,n ≥ 1 such that α(u)n + β(u)m = 0 for all u ∈ L. It is shown that, under these assumptions, either L is central or R ⊆ M2(C) (where M2(C) is the ring of 2 × 2 matrices over C), L is commutative, and u2 ∈ Z for all u ∈ L. In particular, if L =[R, R], then R is commutative.