Abelian Lie group

Let $G$ be a Lie group .

Definition. $G$ is abelian if its multiplication is commutative:

Equivalently, the commutator subgroup $[G,G]$ is trivial, or (for connected $G$) its Lie algebra $\mathfrak{g}=\mathrm{Lie}(G)$ is an abelian Lie algebra .

Motivation. For abelian $G$, the Baker–Campbell–Hausdorff formula collapses to ordinary addition in the Lie algebra: in exponential coordinates, local multiplication is just $X+Y$. This is one reason connected abelian Lie groups admit an explicit classification (see structure of connected abelian Lie groups ).

Remark. Any quotient of an abelian Lie group by a closed subgroup is again abelian, and its Lie algebra is a quotient Lie algebra of $\mathfrak{g}$.