As in Weyl's treatment of compact Lie groups, Gelfand-Naimark for $GL_n(\mathbb C)$, Harish-Chandra's treatment of characters of reductive Lie groups $G$: the regular semi-simple elements $g$ form a set of full measure in the group, and the centralizer $Z(g)$ of regular semi-simple $g$ includes a maximal torus. For compact $G$, all these centralizers are conjugate, so $G/Z(g)$ is isomorphic as $G$-subspace (with conjugation) of full measure in $G$, and Weyl's character formula and dimension formula fall out. For complex reductive (following Gelfand-Naimark and Harish-Chandra) there's again a single conjugacy class. For real reductive, Hirai and Harish-Chandra had to worry about "patching conditions" for characters at the boundaries/interstices between the finitely-many conjugacy classes.
(A similar structural thing happens in p-adic reductive groups...)