Let $G$ be a connected Lie group, $[G,G]$ its derived subgroup (i.e. subgroup generated by commutators of elements in $G$). Here is my question:
Is $[G,G]$ always a Lie subgroup, meaning a does it have the structure of a Lie group such that the inclusion map $[G,G]\rightarrow G$ is an immersion? (i.e. I do not require $[G,G]$ to be a submanifold with respect to the induced topology) If so, why?