Commutativity and Maximal Tori in Connected, Compact Lie Groups

Question

Let $G$ be a path-connected, compact Lie Group. Let $x \in G$ and let $T_x \subset G$ denote the union of all the maximal tori in $G$ that contain $x$.

Question: Is it true that if $y \notin T_x$, then $xy \neq yx$?

Appreciate everyone's help! Thanks!

Answer 1

No, for SO(3) where x, y are distinct commuting involutions.