I discovered a very beautiful proof here that if $G$ is a compact connected Lie group and $T$ a maximal torus then any element $x \in G$ is conjugated to an element of $T$. But for conclude one needs to know the cohomology of $G/T$ (more precisely one needs to know that $G/T$ has a CW-decomposition with only even cells). Apparently this can be done using Morse theory. But I couldn't find anything about this topic in the notes.
I would be very interested to know more about it, where could I find such proof ?