CS-Baer correspondence

Definition

The CS-Baer correspondence (short for central series Baer correspondence) is a generalization of the Baer correspondence that operates as follows:

CS-Baer Lie group $G$ along with central subgroup $H$ in which every element of $[G,G]$ has a unique square root $\leftrightarrow$ CS-Baer Lie ring $L$ along with central subring $M$ in which every element of $[L,L]$ has a unique half

Here a CS-Baer Lie group $G$ is simply a group of nilpotency class two that admits a central subgroup $H$ as described above, and CS-Baer Lie ring $L$ is simply a Lie ring of nilpotency class two that admits a central subring $M$ as described above.

A special subcorrespondence is where $H=Z(G)$ and where $M=Z(L)$. These groups and Lie rings correspond with each other.

From group to Lie ring

Suppose $G$ is a LCS-Baer Lie group and $H$ is a central subgroup of $G$ containing $[G,G]$ such that every element of $[G,G]$ has a unique square root in $H$. Then, $G$ has

Lie ring operation that we need to define	Definition in terms of the group operations	Further comments
Addition, i.e., define $x+y$ for $x,y\in G$	$x+y:={\frac {xy}{\sqrt {[x,y]}}}$	Since $[x,y]\in [G,G]$, it has a unique square root in $H$. We denote this unique square root as ${\sqrt {[x,y]}}$. Note that $H$ is in the center, hence the element ${\sqrt {[x,y]}}$ is central. Thus, to divide by it, we do not need to specify whether we are dividing on the left or on the right.
Identity element for addition, denoted $0$.	Same as identity element for group multiplication, denoted $e$ or $1$.	This automatically follows from the way addition is defined.
Additive inverse, i.e., define $-x$ for $x\in G$.	Same as $x^{-1}$, i.e., the multiplicative inverse in the group.	This automatically follows from the way addition is defined.
Lie bracket, i.e., the $[,]$ map in the Lie ring.	Same as the group commutator $[x,y]=xyx^{-1}y^{-1}$.