Exponential of sum of commuting matrices is product of exponentials

Statement
Suppose $$K$$ is a field. Suppose $$A,B$$ are commuting $$n \times n$$ matrices over $$K$$ such that the matrix exponential is defined for both $$A$$ and $$B$$. Then, the matrix exponential is also defined for the sum of matrices $$\! A + B$$, and:

$$\! \exp(A + B) =\exp A \exp B = \exp B \exp A$$

Related facts

 * Exponential map restricts to homomorphism from abelian subalgebras to abelian subgroups
 * Matrix exponential commutes with conjugation
 * Exponential map commutes with adjoint action