Inverse Baker-Campbell-Hausdorff formula for nilpotency class two

Statement
Consider the Baer correspondence, i.e., the Lazard correspondence restricted to the case of nilpotency class two. Under this correspondence, we have the following formula for $$\exp(\log(X) + \log(Y))$$ (which, if we use the direct identification, is just $$X + Y$$):

$$\exp(\log X + \log Y) = \frac{XY}{\sqrt{[X,Y]}}$$

where $$\sqrt{[X,Y]}$$ denotes the square root of the commutator of $$X,Y$$ in the group. Note that the square root is a well defined and unique element of the center of the group, because one of the assumptions needed for the Baer correspondence to apply is unique 2-divisibility. Thus, we can write this quotient notation without ambiguity, i.e., it is not necessary to explicitly specify whether we are taking the quotient on the left or the right.

Facts used

 * 1) uses::Baker-Campbell-Hausdorff formula for nilpotency class two
 * 2) uses::Formula for group commutator in terms of Lie bracket for nilpotency class two
 * 3) Two elements have Lie bracket zero iff they commute in the group, and if so, their Lie ring sum equals their group product
 * 4) The $$n$$-fold multiple of an element in the Lie ring is the same as its $$n^{th}$$ power in the group.

Proof
For the proof, we adopt the convention of directly identifying the elements of the group and the Lie ring. In other words, what we want to show is that:

$$X + Y = \frac{XY}{\sqrt{[X,Y]_{\mbox{grp}}}}$$

The proof proceeds as follows.

By Fact (1), we already have:

$$XY = X + Y + \frac{1}{2}[X,Y]_{\mbox{Lie}}$$

By Fact (2), we also have:

$$[X,Y]_{\mbox{Lie}} = [X,Y]_{\mbox{grp}}$$

Combining,we get:

$$XY = X + Y + \frac{1}{2}[X,Y]_{\mbox{grp}}$$

By the unique 2-divisibility condition and Fact (4), we obtain that:

$$XY = X + Y + \sqrt{[X,Y}_{\mbox{grp}}$$

By Fact (3), and the fact that the element $$\sqrt{[X,Y}_{\mbox{grp}}$$ is central, we obtain that:

$$XY = (X + Y)(\sqrt{[X,Y}_{\mbox{grp}})$$

Rearranging, we get the desired result:

$$X + Y = \frac{XY}{\sqrt{[X,Y]_{\mbox{grp}}}}$$