Class three Lazard correspondence

Definition
Below is discussed a subcorrespondence of the Lazard correspondence:

$$\{ 2,3 \}$$-powered groups where any subset of size at most three generates a nilpotent subgroup of class at most three $$\leftrightarrow$$ $$\{2, 3\}$$-powered Lie rings where any subset of size at most three generates a nilpotent subring of class at most three

Here, a powered group for a set of primes is a group where every element has a unique $$p^{th}$$ root for every prime $$p$$ in that set.

For any fixed odd prime number $$p > 3$$, this restricts to a correspondence:

$$p$$-groups of nilpotency class at most three $$\leftrightarrow$$ $$p$$-Lie rings of nilpotency class at most three

From group to Lie ring
For proof that this construction works, refer: Proof of class three Lazard correspondence from group to Lie ring

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.

From Lie ring to group
For proof that this construction works, refer Proof of class three Lazard correspondence from Lie ring to group

Suppose $$L$$ is a $$\{2,3\}$$-powered Lie ring of nilpotency class three, with addition denoted $$+$$ and the Lie bracket denoted $$[ \, \ ]$$. We can give $$L$$ the structure of a group of nilpotency class three as follows: