On Lie ideals with generalized derivations

Let $R$ be a prime ring with characteristic different from 2, let $U$ be a nonzero Lie ideal of $R$, and let $f$ be a generalized derivation associated with $d$. We prove the following results: (i) If $a\in R$ and $[a,f(U)]=0$ then $a\in Z$ or $d(a)=0$ or $U\subset Z$; (ii) If $f^2(U)=0$ then $U\subset Z$; (iii) If $u^2\in U$ for all $u\in U$ and $f$ acts as a homomorphism or antihomomorphism on $U$ then either $d=0$ or $U\subset Z$.