Exponential map restricts to homomorphism from abelian subalgebras to abelian subgroups

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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.

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