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Let $G$ be a simple algebraic group over an algebraically closed field ofcharacteristic $p>0$ and suppose that $p$ is a very good prime for $G$. Weprove that any maximal Lie subalgebra $M$ of $\mathfrak{g} = {\rm Lie}(G)$ with${\rm rad}(M) \ne 0$ has the form $M = {\rm Lie}(P)$ for some maximal parabolicsubgroup $P$ of $G$. We show that the assumption on $p$ is necessary byproviding a counterexample for groups type ${\rm E}_8$ over fields ofcharacteristic $5$. Our arguments rely on the main results and methods of theclassification theory of finite dimensional simple Lie algebras over fieldsprime characteristic.

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