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