# 2-division-free Baker-Campbell-Hausdorff formula

## Definition

The 2-division-free Baker-Campbell-Hausdorff formula is a variant of the Baker-Campbell-Hausdorff formula where the Lie brackets are replaced by an arbitrary binary operation and the coefficient for any product of $n$ terms is $2^{n-1}$ times its coefficient in the original Baker-Campbell-Hausdorff formula. In particular, it turns out, due to an observation of Lazard about denominators of these formulas, that the new coefficients do not have any power of 2 in their denominators.

## Particular cases

In the case that we have a nilpotent group or equivalently a nilpotent Lie ring, the Baker-Campbell-Hausdorff formula terminates in finitely many steps, because all terms that involve more than a given number of Lie bracket iterations vanish. Below are the formulas for small values of nilpotency class.

For class $c$, the formula makes sense over any field or ring where all the primes (other than 2) less than or equal to $c$ are invertible.

Note that in the abelian case, the exponential map is a homomorphism, and hence the Baker-Campbell-Hausdorff formula just gives $X + Y$.

Nilpotency class Original formula in terms of Lie bracket New formula in terms of $*$ Primes required to be invertible
1 $X + Y$ $\! X + Y$ no condition
2 $X + Y + \frac{1}{2}[X,Y]$ $\! X + Y + (X * Y)$ no condition
3 $X + Y + \frac{1}{2}[X,Y] + \frac{1}{12}[X,[X,Y]] - \frac{1}{12}[Y,[X,Y]]$ $\! X + Y + (X * Y) + \frac{1}{3}((X * (X * Y)) - (Y * (X * Y)))$ 3 only
4 $X + Y + \frac{1}{2}[X,Y] + \frac{1}{12}[X,[X,Y]] - \frac{1}{12}[Y,[X,Y]] - \frac{1}{24}[Y,[X,[X,Y]]]$ $\! X + Y + (X * Y) + \frac{1}{3}((X * (X * Y)) - (Y * (X * Y))) - \frac{1}{3}(Y * (X * (X * Y)))$ 3 only