Lie ideals and derivations of exceptional prime rings

A prime ring $R$ with extended centroid $C$ is said to be exceptional if both $\text{\rm char}\,R=2$ and $\dim_CRC=4$. Herstein characterized additive subgroups $A$ of a nonexceptional simple ring $R$ satisfying $\big[A, [R, R]\big]\subseteq A$. In 1972 Lanski and Montgomery extended Herstein's theorem to nonexceptional prime rings. In the paper we first extend Herstein's theorem to arbitrary simple rings. For the prime case, let $R$ be an exceptional prime ring with center $Z(R)$. It is proved that if $A$ is a noncentral additive subgroup of $R$ satisfying $\big[A, L\big]\subseteq A$ for some nonabelian Lie ideal $L$ of $R$, then $\beta Z(R)\subseteq A$ for some nonzero $\beta\in Z(R)$, and either $AC=Ca+C$ for some $a\in A\setminus Z(R)$ with $a^2\in Z(R)$ or $[RC, RC]\subseteq AC$. Secondly, we study certain generalized linear identities satisfied by Lie ideals and then completely characterize derivations $\delta, d$ of $R$ satisfying $\delta d(L)\subseteq Z(R)$ for $L$ a Lie ideal of $R$.