The Campbell-Baker-Hausdorff formula says the following $$e^A e^B=e^{A+B+\frac{1}{2}[A,B]+\dots}$$
I am reading a text that says the following
The Campbell-Baker-Hausdorff formula implies that $e^{A+B}=e^A e^B e^{-\frac{1}{2}[A,B]+\dots}$
How does the first formula imply the second? The whole point of this concept is that we can't simply separate $e^{A+B+\frac{1}{2}[A,B]\dots}$ into $e^{A+B}e^{\frac{1}{2}[A,B]+\dots}$
$e^{A+B} = e^A e^B D$ where $D = e^{-B}e^{-A}e^{A+B}$. Now by CBH applied to $-A$ and $A+B$, $$\eqalign{e^{-A} e^{A+B} &= \exp\left(B + \frac{1}{2} [-A,A+B] + \ldots \right)\cr &= \exp\left(B - \frac{1}{2}[A,B] + \ldots\right)}$$ and then using CBH again with $-B$ and $B - \frac{1}{2}[A,B]+\ldots$, $$ \eqalign{D = &\exp \left(-\frac{1}{2} [A,B] + \frac{1}{2} [-B, B - \frac{1}{2} [A,B] + \ldots]\right)\cr & = \exp\left( - \frac{1}{2} [A,B] + \ldots\right) } $$ (here anything with a product of three or more $A$'s and $B$'s goes in the $\ldots$).