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

Let G G be an exceptional simple algebraic group over an algebraically closed field k k and suppose that p = char ( k ) p={\operatorname {char}}(k) is a good prime for G G . In this paper we classify the maximal Lie subalgebras m \mathfrak {m} of the Lie algebra g = Lie ⁡ ( G ) \mathfrak {g}=\operatorname {Lie}(G) . Specifically, we show that either m = Lie ⁡ ( M ) \mathfrak {m}=\operatorname {Lie}(M) for some maximal connected subgroup M M of G G , or m \mathfrak {m} is a maximal Witt subalgebra of g \mathfrak {g} , or m \mathfrak {m} is a maximal exotic semidirect product. The conjugacy classes of maximal connected subgroups of G G are known thanks to the work of Seitz, Testerman, and Liebeck–Seitz. All maximal Witt subalgebras of g \mathfrak {g} are G G -conjugate and they occur when G G is not of type E 6 {\mathrm {E}}_6 and p − 1 p-1 coincides with the Coxeter number of G G . We show that there are two conjugacy classes of maximal exotic semidirect products in g \mathfrak {g} , one in characteristic 5 5 and one in characteristic 7 7 , and both occur when G G is a group of type E 7 {\mathrm {E}}_7 .