Maximal subalgebras of Cartan type in the exceptional Lie algebras

In this paper, we initiate the study of the maximal subalgebras of exceptional simple classical Lie algebras (mathfrak {g}) over algebraically closed fields k of positive characteristic p, such that the prime characteristic is good for (mathfrak {g}). We deal with what is surely the most unnatural case; that is, where the maximal subalgebra in question is a simple subalgebra of non-classical type. We show that only the first Witt algebra can occur as a subalgebra of (mathfrak {g}) and give an explicit classification of when it is maximal in (mathfrak {g}).