Let R be a 2-torsion free prime ring, U a nonzero Lie ideal of R such that $u^2;{in};U$ for all $u;{in};U$. In the present paper, it is proved that if d is a nonzero derivation and [[d(u), u], u] = 0 for all $u;{in};U$, then $U;{subseteq};Z(R)$. Moreover, suppose that $d_1$, $d_2$, $d_3$ are nonzero derivations of R such that $d_3(y)d_1(x);=;d_2(x)d_3(y)$ for all x, $y;{in};U$, then $U;{subseteq};Z(R)$. Finally, some examples are given to demonstrate that the restrictions imposed on the hypothesis of the above results are not superfluous.