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

From Groupprops

## Contents

## Statement

### Statement for a real Lie group

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

### 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