Springer Isomorphisms In Characteristic $p$

Let $G$ be a simple algebraic group over an algebraically closed field of characteristic $p$, and assume that $p$ is a very good prime for $G$. Let $P$ be a parabolic subgroup whose unipotent radical $U_P$ has nilpotence class less than $p$. We show that there exists a particularly nice Springer isomorphism for $G$ which restricts to a certain canonical isomorphism $$\text{Lie}$(U_P) \xrightarrow{\sim} U_P$ defined by J.-P. Serre. This answers a question raised both by G. McNinch in \cite{M2}, and by J. Carlson $\mathit{\text{et.}}\mathit{\text{al}}$ in \cite{CLN}. For the groups $S{L}_n,S{O}_n$, and $S{p}_{2n}$, viewed in the usual way as subgroups of $G{L}_n$ or $G{L}_{2n}$, such a Springer isomorphism can be given explicitly by the Artin-Hasse exponential series.