Commutator subgroup of a Lie group

Let $G$ be a Lie group. Its commutator subgroup (or derived subgroup) is the abstract subgroup

[G,G]=\langle ghg^{-1}h^{-1}:g,h\in G\rangle.

It is the smallest normal subgroup $N\triangleleft G$ for which $G/N$ is abelian; equivalently, $G/[G,G]$ is the algebraic abelianization of $G$ .

Remarks

If $G$ is connected and $\mathfrak g=\operatorname{Lie}(G)$ , then $[G,G]$ has a natural immersed Lie-subgroup structure and its Lie algebra is the derived subalgebra $[\mathfrak g,\mathfrak g]$ . Connectedness matters: components of a disconnected group can contribute additional commutators.

In general, $[G,G]$ need not be closed. The quotient by its closure $\overline{[G,G]}$ is the maximal Hausdorff abelian quotient of $G$ ; the closure is a Lie subgroup by the closed subgroup theorem.

The group $G$ is abelian precisely when $[G,G]$ is trivial.