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