Classification of the maximal subalgebras of exceptional Lie algebras over fields of good characteristic

Let $G$ be an exceptional simple algebraic group over an algebraically closed field $k$ and suppose that the characteristic $p$ of $k$ is a good prime for $G$. In this paper we classify the maximal Lie subalgebras $\mathfrak{m}$ of the Lie algebra $\mathfrak{g}=\mathrm{L}\mathrm{i}\mathrm{e}(G)$. Specifically, we show that one of the following holds: $\mathfrak{m}=\mathrm{L}\mathrm{i}\mathrm{e}(M)$ for some maximal connected subgroup $M$ of $G$, or $\mathfrak{m}$ is a maximal Witt subalgebra of $\mathfrak{g}$, or $\mathfrak{m}$ is a maximal $\it{\mbox{exotic semidirect product}}$. The conjugacy classes of maximal connected subgroups of G are known thanks to the work of Seitz, Testerman and Liebeck–Seitz. All maximal Witt subalgebras of $\mathfrak{g}$ are $G$-conjugate and they occur when $G$ is not of type ${\mathrm{E}}_6$ and $p-1$ coincides with the Coxeter number of $G$. We show that there are two conjugacy classes of maximal exotic semidirect products in $\mathfrak{g}$, one in characteristic $5$ and one in characteristic $7$, and both occur when $G$ is a group of type ${\mathrm{E}}_7$.