A modular analogue of Morozov's theorem on maximal subalgebras of simple Lie algebras

Let $G$ be a simple algebraic group over an algebraically closed field of characteristic $p>0$ and suppose that $p$ is a very good prime for $G$. We prove that any maximal Lie subalgebra $M$ of $\mathfrak{g}=\mathrm{L}\mathrm{i}\mathrm{e}(G)$ with $\mathrm{r}\mathrm{a}\mathrm{d}(M)\mathrm{\ne }0$ has the form $M=\mathrm{L}\mathrm{i}\mathrm{e}(P)$ for some maximal parabolic subgroup $P$ of $G$. We show that the assumption on $p$ is necessary by providing a counterexample for groups type ${\mathrm{E}}_8$ over fields of characteristic $5$. Our arguments rely on the main results and methods of the classification theory of finite dimensional simple Lie algebras over fields prime characteristic.