I've read in a few places (e.g., here), that one can prove Cartan's result on the surjectivity of the exponential map on compact Lie (throughout, assumed to be connected) groups using Lefschetz's fixed point theorem. Now, the most direct proof of the theorem I know of is the usual argument with Hopf-Rinow, but I don't see how one could create a direct proof of Cartan's result using Lefschetz's. One indirect proof is possible: first use the fixed point theorem to prove the maximal torus conjugacy theorem and from there obtain the surjectivity of exponentials for compact lie groups. However, this approach is far from direct.
Is there a direct proof of Cartans's theorem using Lefschetz's?