Classification of connected abelian Lie groups ($G\cong {\mathbb{R}}^n/\Lambda \cong {\mathbb{T}}^r\times {\mathbb{R}}^{n-r}$)

For a connected abelian Lie group, the Lie exponential map is an additive group homomorphism and a local diffeomorphism at zero. Its image is an open subgroup, hence the whole connected group. Its kernel is a discrete additive subgroup of a real vector space. A basis of that kernel identifies it with ${\mathbb{Z}}^r$ inside ${\mathbb{R}}^n$, giving the displayed classification. The compact factors record periods of one-parameter subgroups.