Commutator subgroup of a Lie group

Let $G$ be a Lie group .

Definition. The commutator subgroup (or derived subgroup) $[G,G]$ is the subgroup generated by all commutators

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

Lie-theoretic relation. If $\mathfrak{g}=\mathrm{Lie}(G)$, then the Lie algebra of the identity component of $[G,G]$ is the derived subalgebra $[\mathfrak{g},\mathfrak{g}]$. This makes $[G,G]$ the global counterpart of “taking brackets” in $\mathfrak{g}$.

Topological subtlety. In general, $[G,G]$ need not be closed. When forming a smooth quotient, one often replaces it by its closure $\overline{[G,G]}$, which is a closed subgroup and hence a Lie subgroup by the closed subgroup theorem .

Context. $G$ is abelian precisely when $[G,G]$ is trivial, and the size of $[G,G]$ controls how far $G$ is from being commutative.