Exponential map commutes with adjoint action

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.