An abelian Lie group has commutative multiplication. Its Lie algebra is an abelian Lie algebra, and its Lie exponential map is a group homomorphism from the additive Lie algebra. Conversely, an additive Lie exponential map forces the identity component of a Lie group to be abelian, but does not force all disconnected components to commute.
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 inside , giving the displayed classification. The compact factors record periods of one-parameter subgroups.
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An **Abelian Lie group** is a type of Lie group in which the group operation is commutative. This means that for any two elements \( g \) and \( h \) in the group \( G \), the following property holds: \[ g \cdot h = h \cdot g \] where \( \cdot \) represents the group operation.