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The BCH is given by

$e^{SX}Ye^{-sX}=Y+s[X,Y]+\frac{s^2}{2}[X,[X,Y]]+\text{...}$

How do you get to the Zassenhaus formula?

$e^{X+Y}=e^{sX}e^{sY}e^{-1/2s^2[X,Y]}\text{...}$

Cameron
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    In what you write, this alleged "BCH" formula is not what's usually called BCH formula. In your formula $X$ belongs to the Lie algebra and $Y$ to the Lie group, so $[X,Y]$ is senseless. Also the left-hand term is in the Lie group (multiplication) and the right-hand term in the Lie algebra (addition). – YCor Dec 20 '17 at 21:17

1 Answers1

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The Zassenhaus formula is a kind of dual of the Baker–Campbell–Hausdorff formula. On the Wikipedia page for BCH, where you find the correct BCH formula, the Zassenhaus formula is also given. It is explained that "these exponents, $C_n$ in $\exp(–tX) exp(t(X+Y)) = \prod_n \exp(t_n C_n)$, follow recursively by application of the above BCH expansion." For more details see the article by Nishimura here, section $6$ and $7$.

Dietrich Burde
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