Let $G$ be a compact connected Lie group, $T$ be some maximal torus in $G$ (that is, inclusion-maximal connected abelian subgroup). Then the union of tori $gTg^{-1}$, $g\in G$, is the whole $G$. This is well-known (4.21 in Adams book). My question is rather methodological: is there any proof without use of algebraic topology? Adams presents A. Weil's proof, which uses some kind of Lefschetz fixed points theorem. 
(Yes, I am sorry but my motivation is mostly that I teach second year students this stuff.) 



maximal tori cover compact Lie group
There are proofs that avoid algebraic topology: see for example Chapter 16 in Bump's *Lie Groups* or IV.5 in Knapp's *Lie Groups Beyond an Introduction*.


