Let $G \le GL(n,\mathbb C)$ be a matrix Lie group, regarded as an embedded real submanifold of $M(n,\mathbb C)$, and let $\mathfrak g=T_I G\subset M(n,\mathbb C)$ be its [Lie algebra](/page/Lie%20Algebra). Let $\exp:\mathfrak g\to G$ be the restriction of the matrix exponential. For $A\in G$, let $R_A:G\to G$ denote the smooth right translation map $R_A(B)=BA$. For $X\in\mathfrak g$, define $\operatorname{ad}_X:\mathfrak g\to\mathfrak g$ by $\operatorname{ad}_X(Z)=[X,Z]=XZ-ZX$. Then, for every $X,Y\in\mathfrak g$, the series $\sum_{k=0}^{\infty}(\operatorname{ad}_X)^k(Y)/(k+1)!$ converges in the finite-dimensional [vector space](/page/Vector%20Space) $\mathfrak g$, and

\begin{align*} (d\exp)_X(Y)=(dR_{\exp X})_I\left(\sum_{k=0}^{\infty}\frac{(\operatorname{ad}_X)^k(Y)}{(k+1)!}\right). \end{align*}

Equivalently,

\begin{align*} (dR_{\exp(-X)})_{\exp X}\bigl((d\exp)_X(Y)\bigr)=\sum_{k=0}^{\infty}\frac{(\operatorname{ad}_X)^k(Y)}{(k+1)!}. \end{align*}