Baer correspondence up to isoclinism

Definition
The Baer correspondence up to isoclinism is a correspondence defined as follows:

Equivalence classes under isoclinism of groups of nilpotency class at most two $$\leftrightarrow$$ Equivalence classes under isoclinism of Lie rings of nilpotency class at most two

The correspondence is as follows: A Lie ring $$L$$ is identified with a group $$G$$ via a pair of isomorphisms:


 * An isomorphism $$\zeta$$ between the additive group of $$L/Z(L)$$ and the inner automorphism group $$G/Z(G)$$, and
 * An isomorphism $$\phi$$ between the additive group of $$[L,L]$$ and the derived subgroup $$[G,G]$$

such that for $$x,y \in L$$, with images $$\overline{x},\overline{y}$$ mod $$Z(L)$$, we have:

$$\phi([x,y]) = [\zeta(\overline{x}),\zeta(\overline{y})]$$

where the bracket on the left is the Lie bracket and the bracket on the right is the group commutator, well defined because the group commutator in $$G$$ of two elements depends only on their cosets mod $$Z(G)$$.