On medium-rank Lie primitive and maximal subgroups of exceptional groups of Lie type

We study embeddings of groups of Lie type H in characteristic p into exceptional algebraic groups G of the same characteristic. We exclude the case where
H is of type PSL₂. A subgroup of G is Lie primitive if it is not contained in any proper, positive-dimensional subgroup of G.

With a few possible exceptions, we prove that there are no Lie primitive subgroups H in G, with the conditions on H and G given above. The exceptions are for H one of PSL₃(3), PSU₃(3), PSL₃(4), PSU₃(4), PSU₃(8), PSU₄(2), PSp₄(2)'
and ²B₂(8), and G of type E₈. No examples are known of such Lie primitive embeddings.

We prove a slightly stronger result, including stability under automorphisms of G. This has the consequence that, with the same exceptions, any almost simple group with socle H, that is maximal inside an almost simple exceptional group of
Lie type F₄, E₆, ²E₆, E₇ and E₈, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup inside the algebraic group.

The proof uses a combination of representation-theoretic, algebraic group-theoretic, and computational means.