  We study reductive subgroups $H$ of a reductive linear algebraic group $G$ --
possibly non-connected -- such that $H$ contains a regular unipotent element of
$G$. We show that under suitable hypotheses, such subgroups are $G$-irreducible
in the sense of Serre. This generalizes results of Malle, Testerman and
Zalesski. We obtain analogous results for Lie algebras and for finite groups of
Lie type. Our proofs are short, conceptual and uniform.
