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 $Z(G)$.