Lie group, derived

The commutator subgroup of a Lie group. For any Lie group $G$ its derived Lie group $[G,G]$ is a normal (not necessarily closed) Lie subgroup of $G$. The corresponding ideal of the Lie algebra $\mathfrak g$ of the group $G$ coincides with the commutator algebra $[\mathfrak g,\mathfrak g]$ (also called the derived Lie algebra of $\mathfrak g$). The commutator subgroup of a simply-connected (or connected linear) Lie group $G$ is always closed in $G$.

References

[1]	C. Chevalley, "Theory of Lie groups" , 1 , Princeton Univ. Press (1946)