Generic irreducibilty of Laplace eigenspaces on certain compact Lie groups

If $G$ is a compact Lie group endowed with a left invariant metric $g$, then $G$ acts via pullback by isometries on each eigenspace of the associated Laplace operator $\Delta_g$. We establish algebraic criteria for the existence of left invariant metrics $g$ on $G$ such that each eigenspace of $\Delta_g$, regarded as the real vector space of the corresponding real eigenfunctions, is irreducible under the action of $G$. We prove that generic left invariant metrics on the Lie groups $G=\operatorname{SU}(2)\times\ldots\times\operatorname{SU}(2)\times T$, where $T$ is a (possibly trivial) torus, have the property just described. The same holds for quotients of such groups $G$ by discrete central subgroups. In particular, it also holds for $\operatorname{SO}(3)$, $\operatorname{U}(2)$, $\operatorname{SO}(4)$.