Let $G \le GL(n,\mathbb C)$ be a matrix Lie group with identity element $I$ and [Lie algebra](/page/Lie%20Algebra) $\mathfrak g$. There exist neighbourhoods $U_X,U_Y\subset\mathfrak g$ of $0$ and a logarithm chart $\log:V\to U$ from an open neighbourhood $V\subset G$ of $I$ to an open neighbourhood $U\subset\mathfrak g$ of $0$ such that for every $X\in U_X$ and $Y\in U_Y$, the path

\begin{align*} \alpha:[0,1]\to G,\qquad t\mapsto \exp X\exp(tY) \end{align*}

has image contained in $V$. Define

\begin{align*} Z:[0,1]\to\mathfrak g,\qquad t\mapsto \log(\exp X\exp(tY)). \end{align*}

Then $Z$ is smooth and satisfies, for every $t\in[0,1]$,

\begin{align*} \frac{dZ}{dt}(t)=\frac{\operatorname{ad}_{Z(t)}}{1-e^{-\operatorname{ad}_{Z(t)}}}(Y). \end{align*}

Here $\operatorname{ad}_{Z(t)}:\mathfrak g\to\mathfrak g$ is the [linear map](/page/Linear%20Map) $W\mapsto[Z(t),W]$, and the quotient denotes the convergent operator [power series](/page/Power%20Series) inverse to

\begin{align*} \frac{1-e^{-\operatorname{ad}_{Z(t)}}}{\operatorname{ad}_{Z(t)}}. \end{align*}