Suppose $G$ is a closed subgroup of $SU(d)$, $d>1$, and let $\rho$ be a $d$-dimensional special unitary representation of $G$. Suppose that if a matrix $A$ commutes with all of $\rho(G)$ for all $g\in G$, then $A=cI$. Does it then hold that $\rho$ is an irreducible representation of $G$?
Schur's first lemma states that if $\rho$ is irreducible, then if a matrix $A$ commutes with all of $\rho(G)$ for all $g\in G$, then $A=cI$. I'm asking if the converse of this statement holds. If it doesn't hold in general, does it hold in any special cases? I'm particularly interested in the case where $d=3$ and $G$ is either finite or continuous and closed.