I'm studying QM out of Sakurai's book and on page 95 (equation 2.3.47), he states the Baker-Hausdorff Lemma without proof. I've scoured the internet in search of an elementary proof (i.e. one that does not rely on lie algebras, etc.), but have come up empty. I was wondering if anyone knew of a place I could find a proof online.

Check out problems 3.3 & 3.4 in the text on Quantum Mechanics by Ballentine. They are solved in the appendix D. It should be a start to the general formula, indeed without involving topological issues.