  Let $G$ be a simple connected complex Lie group. The additive eigencone of
$G$ is a polyhedral cone containing the set of solutions to the additive
eigenvalue problem, a generalization of the Hermitian eigenvalue problem. The
additive eigencone is functorial, and for certain subgroups satisfies a
stronger functoriality property: the eigencone of the subgroup is determined by
the inequalities of the larger eigencone. Belkale and Kumar first studied this
property for subgroups invariant under a diagram automorphism of $G$. We study
a new class of subgroups arising from centralizers of torus elements that have
the strong eigencone functoriality property.
