I would be grateful if someone could provide a reference/proof of the following fact (or give a counterexample if I've misunderstood and it's false!)
Let $G$ be a reductive group over a field $F$ (in practice I really only care about local and global fields). Suppose that $G$ contains an elliptic maximal $F$-torus (i.e. a maximal torus defined over $F$ that is anisotropic modulo $Z(G)$ ). Suppose that $\gamma \in G(F)$ is an elliptic element (i.e. the identity component of the center of the identity component of the centralizer of $\gamma$ in $G$ is an elliptic torus).
Then, fact: $\gamma$ is contained in an elliptic maximal torus.
This is alluded to on page 392 of Kottwitz's Stable Trace Formula: Elliptic Singular Terms, but I would appreciate a reference that proves this fact, or at least confirmation that I am not misunderstanding this passage.