Primitive Subalgebras of Exceptional Lie Algebras

The object of this paper is to classify (up to inner automorphism) the primitive, maximal rank, reductive subalgebras of the (complex) exceptional Lie algebras. By primitive we mean that the subalgebras correspond to (possibly disconnected) maximal Lie subgroups. In [3], the corresponding classification for the (complex) classical Lie algebras was completed, as was the classification for the nonreductive, maximal rank, subalgebras of all the simple Lie algebras. Using case by case techniques and some more general results given later, we prove the following theorem: