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 $$X + Y$$.