Exponential map restricts to homomorphism from abelian subalgebras to abelian subgroups

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.

Related facts

 * Baker-Campbell-Hausdorff formula provides a more general version
 * Exponential map commutes with adjoint action, and a matrix version of this is matrix exponential commutes with conjugation