Prove Taylor expansion $(\mathrm{exp}\,tX)(\mathrm{exp}\,tY)=\mathrm{exp}(t(X+Y)+\frac{1}{2}t^2[X,Y]+o(t^2))$

Question

This is an exercise from John Lee’s book Introduction to Smooth Manifolds, GTM218, chapter 20. It says

For Lie group $G$ and two vector fields $X,Y\in \mathrm{Lie}(G)$, we have $$(\mathrm{exp}\,tX)(\mathrm{exp}\,tY)=\mathrm{exp}(t(X+Y)+\frac{1}{2}t^2[X,Y]+o(t^2))$$ for $t\in\mathbb{R}$ near origin.

I can explain the first term $t(X+Y)$ by differentiating $\varphi(t)=\mathrm{exp}^{-1}((\mathrm{exp}\,tX)(\mathrm{exp}\,tY))$ at $t=0$, but have trouble calculating higher differentials. I know it’s a special form of the Baker–Campbell–Hausdorff formula, but it should be solved within the scope of the basic knowledge of Lie group and exponential map.

Answer 1

The expansion can be derived easily when $G$ is a matrix Lie group, since then we have an explicit formula for $\exp$. My question concerns whether it is true for arbitrary Lie groups. Notice the local nature of the problem: the expansion holds when $|t|$ is sufficiently small. Thus we may invoke Lie's theorems which tell us that any Lie group is locally isomorphic to a matrix Lie group. I think this is sufficient to derive the general case. Any comments are appreciated.

Answer 2

Hint: expand both sides to $O(t^2)$ and compare