Exponential map commutes with adjoint action

Statement

Statement for a real Lie group

Suppose $G$ is a real Lie group and ${\mathfrak {g}}$ is its Lie algebra. For $g\in G$ and $x\in {\mathfrak {g}}$, the following is true:

$\exp(\operatorname {Ad} (g)(x))=g\exp(x)g^{-1}$

where $\operatorname {Ad}$ denotes the adjoint action of Lie group on Lie algebra.

Statement for a linear Lie group

For a linear Lie group, the statement reduces to matrix exponential commutes with conjugation.