Let R be a prime ring with center Z and characteristic different from two, U a nonzero Lie ideal of R such that $u^2{in}U$ for all $u{in}U$ and $d$ be a nonzero (${sigma}$, ${ au}$)-derivation of R. We prove the following results: (i) If $[d(u),u]_{{sigma},{ au}}$ = 0 or $[d(u),u]_{{sigma},{ au}}{in}C_{{sigma},{ au}}$ for all $u{in}U$, then $U{subseteq}Z$. (ii) If $a{in}R$ and $[d(u),a]_{{sigma},{ au}}$ = 0 for all $u{in}U$, then $U{subseteq}Z$ or $a{in}Z$. (iii) If $d([u,v])={pm}[u,v]_{{sigma},{ au}}$ for all $u{in}U$, then $U{subseteq}Z$.