Let $\mathbb F\in\{\mathbb R,\mathbb C\}$. Let $\mathfrak g$ be a finite-dimensional [Lie algebra](/page/Lie%20Algebra) over $\mathbb F$, equipped with a norm $\|\cdot\|_{\mathfrak g}$ such that $\|[A,B]\|_{\mathfrak g}\leq \|A\|_{\mathfrak g}\|B\|_{\mathfrak g}$ for all $A,B\in\mathfrak g$. For $X,Y\in\mathfrak g$, define the Dynkin Baker--Campbell--Hausdorff Lie series $\operatorname{BCH}(X,Y)$ by

\begin{align*} \operatorname{BCH}(X,Y)=\sum_{m=1}^{\infty}\frac{(-1)^{m-1}}{m}\sum_{\substack{r_i, s_i\geq 0, r_i+s_i>0}}\frac{C(X^{r_1}Y^{s_1}\cdots X^{r_m}Y^{s_m})}{(r_1+s_1+\cdots+r_m+s_m)r_1!s_1!\cdots r_m!s_m!}. \end{align*}

Here the inner sum is over all $m$-tuples $(r_1,s_1),\dots,(r_m,s_m)$ of non-negative integer pairs with $r_i+s_i>0$, the notation $X^rY^s$ denotes the ordered word consisting of $r$ copies of $X$ followed by $s$ copies of $Y$, and $C(Z_1):=Z_1$ while $C(Z_1,\dots,Z_N):=[Z_1,C(Z_2,\dots,Z_N)]$ for finite words $(Z_1,\dots,Z_N)$ in $\{X,Y\}$. If $\|X\|_{\mathfrak g}+\|Y\|_{\mathfrak g}<\log 2$, then the Dynkin series defining $\operatorname{BCH}(X,Y)$ converges absolutely in $\mathfrak g$.

Moreover, assume the following standard Dynkin BCH exponential theorem for matrix Lie groups: whenever $G\leq GL(n,\mathbb F)$ is a matrix Lie group with Lie algebra $\mathfrak h\subset M(n,\mathbb F)$, $\|\cdot\|_{\mathfrak h}$ is a norm on $\mathfrak h$ satisfying $\|[A,B]\|_{\mathfrak h}\leq \|A\|_{\mathfrak h}\|B\|_{\mathfrak h}$ for all $A,B\in\mathfrak h$, and $A,B\in\mathfrak h$ satisfy $\|A\|_{\mathfrak h}+\|B\|_{\mathfrak h}<\log 2$, the matrix Dynkin series $\operatorname{BCH}_{\mathfrak h}(A,B)$ converges absolutely and satisfies $\exp_G(A)\exp_G(B)=\exp_G(\operatorname{BCH}_{\mathfrak h}(A,B))$. Suppose $n\in\mathbb N$, $G\leq GL(n,\mathbb F)$ is a matrix Lie group with Lie algebra $\mathfrak h\subset M(n,\mathbb F)$, and $\Phi:\mathfrak g\to\mathfrak h$ is a Lie algebra isomorphism. Then, for every $X,Y\in\mathfrak g$ satisfying the same norm bound, $\Phi(\operatorname{BCH}(X,Y))$ is the matrix Dynkin BCH series for $\Phi(X)$ and $\Phi(Y)$, and

\begin{align*} \exp_G(\Phi(X))\exp_G(\Phi(Y))=\exp_G(\Phi(\operatorname{BCH}(X,Y))). \end{align*}