Let $n\in\mathbb N$, let $M(n,\mathbb C)$ denote the complex [vector space](/page/Vector%20Space) of complex $n\times n$ matrices, let $I_n\in M(n,\mathbb C)$ denote the identity matrix, and let $G\le GL(n,\mathbb C)$ be a matrix Lie group with [Lie algebra](/page/Lie%20Algebra) $\mathfrak g\subset M(n,\mathbb C)$, where $\mathfrak g=\{X\in M(n,\mathbb C):\exp(tX)\in G\text{ for every }t\in\mathbb R\}$ under the tangent-space identification $\mathfrak g=T_{I_n}G$. For each $g\in G$, let $C_g:G\to G$ be the conjugation map $C_g(h)=ghg^{-1}$ and let $\operatorname{Ad}_g:=d(C_g)_{I_n}:\mathfrak g\to\mathfrak g$ be the adjoint action. For $X\in\mathfrak g$, let $\operatorname{ad}_X:\mathfrak g\to\mathfrak g$ denote the real-[linear map](/page/Linear%20Map) defined by $\operatorname{ad}_X(Y)=[X,Y]=XY-YX$. Then $\operatorname{Ad}_{\exp X}=\exp(\operatorname{ad}_X)$ as real-linear maps $\mathfrak g\to\mathfrak g$, where $\exp X$ is the matrix exponential and $\exp(\operatorname{ad}_X)$ is the operator exponential on $\operatorname{End}_{\mathbb R}(\mathfrak g)$.