Let R be a prime ring with characteristic not two. U a (${sigma},{ au}$)-left Lie ideal of R and d : R$ ightarrow$R a non-zero derivation. The purpose of this paper is to invesitigate identities satisfied on prime rings. We prove the following results: (1) [d(R),a]=0$Leftrightarrow$d([R,a])=0. (2) if $(R,a)_{{sigma},{ au}}$=0 then $a{in}Z$. (3) if $(R,a)_{{sigma},{ au}}{subset}C_{{sigma},{ au}}$ then $a{in}Z$. (4) if $(U,a){subset}Z$ then $a^2{in}Z;or;{sigma}(u)+{ au}(u){in}Z$, for all $u{in}U$. (5) if $(U,R)_{{sigma},{ au}}{subset}C_{{sigma},{ au}}$ then $U{subset}Z$.