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 p={\operatorname {char}}(k) is a good prime for G. In this paper we classify the maximal Lie subalgebras \mathfrak{m} of the Lie algebra \mathfrak{g}=\operatorname {Lie}(G). Specifically, we show that either \mathfrak{m}=\operatorname {Lie}(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 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.