Formula for group commutator in terms of Lie bracket for nilpotency class two

In log and exp notation
This article describes the formula for group commutator in terms of Lie bracket for groups of nilpotency class two. In particular, it applies to the Baer correspondence.

There are two notions of group commutator, depending on whether we use the left convention or the right convention. Formulas for both cases are presented.

Short version
Consider the Baer correspondence, where we identify a uniquely 2-divisible group of nilpotency class two with its Baer Lie ring. Under this correspondence, we have that:

Group commutator of two elements using left convention = Lie bracket of two elements = Group commutator of two elements using right convention