I'm learning about Lie groups and Lie algebras independently, and I'm trying to show that the commutator subgroup, $H=[G,G]$, of a Lie group, $G$, is a Lie subgroup. My first instinct was to take a look at the top down approach of defining $$\displaystyle H=\bigcap_{i\in I}H_i$$ where $H_i\supset X$, and $X:=\{ghg^{-1}h^{-1}:g,h\in G\}$. It is a standard algebra exercise to show that the intersection is a subgroup, but my difficulty is the fact that in order for it to be a submanifold, the $H_i$ must satisfy certain transversality conditions.
This didn't seem like a natural approach so I decided to look at the bottom up approach. That is I considered defining $H$ as $$H:=\left\{g_1^{a_1}\cdot g_2^{a_2}\cdots g_k^{a_k}:k\in\mathbb{N}, g_k\in X, a_i\in\{-1,1\}\right\}$$
With this approach it is no problem to show this forms a group, but I'm still having some issues with the submanifold part. One thought was to use the fact that $G$ acts on itself via right or left multiplication, and $\Phi_g(h) = hg = h'$ is diffeomorphism from $X$ to some $X_g\subset G$. I'm not sure if this is the right way to go. I would greatly appreciate any hints on where to go next.