I want to solve the following problem (20-4 in Lee's book on smooth manifolds, second edition):
I am quite new to these affairs, and would like you to check my proof.
My attempt at a solution:
In rigor, we should perhaps write
$$
F(X,Y)=\exp(X)|_A\exp|_B(Y)
$$
From Prop. 20.8(e) of the same book, I know that the exponential map restricts to a diffeomorphism from a neighborhood of $0\in \mathfrak{g}$ to a neighborhood of $e\in G$. So there is a neighborhood $U$ of $0$ such that $\exp|_U$ is a diffeomorphism, and therefore both $\exp|_{U\cap A}$ and $\exp|_{U\cap B}$ are also diffeomorphisms.
Take $V=U\cap A\times U\cap B$. Notice that $(0,0)\in V$. Also,
$$
F|_V(X,Y)=\exp|_{U\cap A}(X) \exp|_{U\cap B}(Y)=\\
=\exp|_{U\cap A}(\pi_1(X,Y)) \exp|_{U\cap B}(\pi_2(X,Y))
$$
So $F=(\exp|_{U\cap A}\circ\pi_1)\cdot (\exp|_{U\cap B}\circ\pi_2)$. Being the product of the composition of diffeomorphisms, $F$ is itself a diffeomorphism. $\square$
The differential of $exp$ at $0$ is the identity. This implies that $\partial_XF(X,Y)_{(0,0)}(u)=u$ and $\partial_YF(X,Y)_{(0,0)}(v)=v$. We deduce that $dF_{(0,0)}=Id$. The local inversion theorem implies that $F$ is a local diffeomorphism in a neighborhood of $(0,0)$.
For your argument $dim(U\cap A)<dim G$, so $exp_{\mid U\cap A}$ cannot be a diffeomorphism.