Inverse Baker-Campbell-Hausdorff formula

Statement

The inverse Baker-Campbell-Hausdorff formula is a formula that tries to express:

$\!\exp(\log(X)+\log(Y))$

in terms of $X$ and $Y$. Here $X$ and $Y$ are elements of a Lie group (or analogue thereof), $\log$ is the logarithm map to its Lie ring or Lie algebra, addition happens in the Lie ring, and $\exp$ is the exponential map.

Note that there is a related inverse formula which is the formula for Lie bracket in terms of group commutator.

Particular cases

Nilpotency class	Interpretation	Formula	Primes $p$ for which $p$ should be coprime to orders of elements
1	abelian group, abelian Lie ring	$\!XY$	no condition
2	group of nilpotency class two, Lie ring of nilpotency class two	$\!{\frac {XY}{\sqrt {[X,Y]}}}$	2 only

References

• An effective version of the Lazard correspondence by Serena Cicalo, Willem A. de Graaf and M. R. Vaughan-Lee, Journal of Algebra, ISSN 00218693, Volume 352, Page 430 - 450(Year 2012): ^{[dx.doi.org/10.1016/j.jalgebra.2011.11.031 Official copy (gated)]More info}