I am reading Fulton-Harris Representation Theory and am trying to do Exercise 8.42:
Let $G_0=\exp(\Delta)$, where $\Delta$ is a disk centered at the origin in $\mathfrak g$, and let $H_0=\exp(\Delta\cap\mathfrak h)$. Show that $G_0^{-1}=G_0,H_0^{-1}=H_0$, and $H_0\cdot H_0\cap G_0=H_0$. Use this to show that the subgroup $H$ of $G$ generated by $H_0$ is an immersed Lie subgroup of $G$.
Background: This exercise is used in the book to prove Proposition 8.41:
Let $G$ be a Lie group, $\mathfrak g$ its Lie algebra, and $\mathfrak h\subset\mathfrak g$ a Lie subalgebra. Then the subgroup of $G$ generated by $\exp(\mathfrak h)$ is an immersed subgroup $H$ with tangent space $T_eH=\mathfrak h$.
What I have so far:
I am able to prove $G_0^{-1}=G_0,H_0^{-1}=H_0$, and $H_0\cdot H_0\cap G_0=H_0$. The first two are evident as $\Delta$ is a disk. For the last equality, I understand it as $(H_0\cdot H_0)\cap G_0=H_0$. Assuming $\Delta$ is small enough on which $\exp$ is a diffeomorphism and we can use Baker-Campbell-Hausdorff formula, we get that if $\exp(X)\exp(Y)=\exp(Z)$ where $X,Y\in\Delta\cap\mathfrak h,Z\in\Delta$, then $Z$ is described by the $BCH$ formula, so $Z\in\mathfrak h$ and $\exp(Z)\in H_0$.
The subgroup $H$ generated by $H_0$ is (please correct me if I'm wrong) $$H=\cup_{n\ge 1}H_0^n$$ I am stuck at showing $H$ is an immersed subgroup. Precisely, I think I should show $H\cap G_0=H_0$, so $H_0$ is open in $H$. But I don't know how to show $H_0^n\cap G_0=H_0$ for $n\ge 3$.
Then provided $H\cap G_0=H_0$, we can endow $H$ with a differentiable structure induced by $\exp$ (as $\exp$ is a diffeomorphism on $\Delta$, let $(H_0,\exp^{-1})$ be a chart on $H$). Under such a structure $H$ is a Lie group with tangent space $T_eH=\mathfrak h$.
Maybe we can prove $H\cap G_0=H_0$ but I didn't see it; maybe there is another way to do this exercise. Any help is greatly appreciated!