I am interested in a reference and proof for some version of the following (folklore?) statement:

``Let $G$ be a (semi)simple Lie group (with no compact factors and trivial centre) and let $\Gamma$ be an (irreducible) arithmetic lattice. Then $\mathrm{Comm}_G(\Gamma)$ is a simple group."

I have yet been unable to locate a precise statement or proof of this result. However, it is briefly alluded to on page 2 of this open problem list.