$A_1$-type subgroups containing regular unipotent elements

Let be a simple exceptional algebraic group of adjoint type over an algebraically closed field of characteristic and let be a subgroup of containing a regular unipotent element of . By a theorem of Testerman, is contained in a connected subgroup of of type . In this paper we prove that with two exceptions, itself is contained in such a subgroup (the exceptions arise when or ). This extends earlier work of Seitz and Testerman, who established the containment under some additional conditions on and the embedding of in . We discuss applications of our main result to the study of the subgroup structure of finite groups of Lie type.