Let $G$ be a finite-dimensional real Lie group with identity element $e$, [Lie algebra](/page/Lie%20Algebra) $\mathfrak g=T_eG$, Lie bracket $[\cdot,\cdot]$, and exponential map $\exp:\mathfrak g\to G$. Then there exist an open neighbourhood $U\subset\mathfrak g$ of $0$ and a smooth map $R:U\times U\to\mathfrak g$ such that, for all $X,Y\in U$, the element $\exp X\exp Y\exp(-X)\exp(-Y)$ lies in the image of a local exponential chart at $e$ and satisfies

\begin{align*} \exp X\exp Y\exp(-X)\exp(-Y)=\exp\bigl([X,Y]+R(X,Y)\bigr). \end{align*}

The Taylor expansion of $R$ at $(0,0)$ has no homogeneous terms of total degree less than $3$. Equivalently,

\begin{align*} \exp X\exp Y\exp(-X)\exp(-Y)=\exp\bigl([X,Y]+O_3(X,Y)\bigr), \end{align*}

where $O_3(X,Y)$ denotes a Lie-algebra-valued local remainder of total degree at least $3$.