Does someone has a reference to a modern proof of the Baker-Campbell-Hausdorff formula?
All proofs I have ever seen are related only to matrix Lie groups / Lie algebras and are not at all geometric (i.e. depend on indices, bases ect.)
By a 'modern' proof I'm thinking of a proof entirely in terms of differential geometry, i.e. in terms of the tangent bundle on the Lie group manifold or even better in terms of jets.
I'll keep the formulation vague on purpose, to higher my chances to get a good reference. I think the question is pretty clear anyway.
Beyond the plain BCH-equation I would like to get a deeper understanding WHY the commutator (and the linear structure of the Lie algebra) is enough to define the group product locally and what is geometrically going on.