Let \(R\) be a prime ring with \(\text{char}(R)\neq 2\), and center \(Z\). A Lie ideal of \(R\) is an additive subgroup \(U\) of \(R\) satisfying \(ur-ru\in U\) for all \(r\in R\) and \(u\in U\). The purpose of the paper is to prove that if \(T\) is an automorphism of \(R\), not the identity on \(U\), so that \(uu^T-u^Tu\in Z\) for each \(u\in U\), then \(U\subseteq Z\). We note that if \(U\not\subset Z\), then the subring generated by \(U\) contains a nonzero ideal of \(R\), so when \(T\) is the identity on \(U\) either \(T=I\) or \(U\subseteq Z\).