Commutativity conditions on derivations and Lie ideals $\sigma$-prime rings

Let $R$ be a 2-torsion free $\sigma$-prime ring, $U$ a nonzero square closed $\sigma$-Lie ideal of $R$ and let $d$ be a derivation of $R$. In this paper it is shown that:

1. If $d$ is centralizing on $U$, then $d=0$ or $U\subseteq Z(R)$.
2. If either $d([x,y])=0$ for all $x,y\in U$, or $[d(x),d(y)]=0$ for all $x,y\in U$ and $d$ commutes with $\sigma$ on $U$, then $d=0$ or $U \subseteq Z®$.