I am curious about this remark in the proof; they have just assumed that a connected Lie group $G$ can be assumed to be a matrix Lie group, and justify it like this:
This is a harmless assumption, given the statement (to be proved in Appendix E) that any finite-dimensional Lie algebra may be embedded in the Lie algebra $\mathfrak{gl}_n\mathbb{R}$: the subgroup of $\mathbf{GL}_n \mathbb{R}$ generated by $\mathrm{exp}(\mathfrak{g})$ will be a group isogenous to G [...]
I cannot see any other way of giving this isogeny than by using the existence of a Baker-Campbell-Hausdorff formula for non-matrix Lie groups. Given this is never mentioned in the book and from what I can tell requires some effort.
I hope I'm missing something, since it would be nice to only have to prove the existence of BCH in the matrix exponential context.