# Exponential map restricts to homomorphism from abelian subalgebras to abelian subgroups

## Statement

### Statement for a real Lie group

Suppose $G$ is a real Lie group and $\mathfrak{g}$ is its Lie algebra. Suppose $\mathfrak{h}$ is an abelian subalgebra of $\mathfrak{g}$. Denote by $H$ the image of $\mathfrak{h}$ under the exponential map. Then, the restriction of the exponential map to $\mathfrak{h}$ gives a homomorphism of groups from $\mathfrak{h}$ (with additive structure) to $H$ (as a multiplicative subgroup of $G$). In particular, $H$ is an abelian subgroup of $G$.

### Statement for a linear Lie group

In this case, the statement is just exponential of sum of commuting matrices is product of exponentials.