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 (Formula presented.) over algebraically closed fields k of positive characteristic p, such that the prime characteristic is good for (Formula presented.). 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 (Formula presented.) and give an explicit classification of when it is maximal in (Formula presented.).