Proof of class three Lazard correspondence from group to Lie ring

This article describes the group to Lie ring direction of the class three Lazard correspondence, which in turn is a special case of the Lazard correspondence

Statement
Suppose $$G$$ is a $$\{ 2,3 \}$$-powered group of nilpotency class at most three. In particular, this means that every element has a unique square root (which we denote by the $$\sqrt{}$$ symbol) and a unique $$12^{th}$$ root (which we denote by the $$\sqrt[12]{}$$ symbol). Note that if $$g \in G$$ has finite order $$m$$, $$m$$ must be relatively prime to both 2 and 3, and further, $$\sqrt{g} = g^{(m + 1)/2}$$ and $$\sqrt[12]{g}$$ is also a power of $$g$$, albeit the exponent expression will depend on the congruence class of $$m$$ mod 12.

The claim is that with these operations, $$G$$ acquires the structure of a Lie ring of nilpotency class at most three.