CS-Baer Lie group

Definition
A CS-Baer Lie group is a group $$G$$ satisfying both the following two conditions:


 * 1) $$G$$ is a defining ingredient::group of nilpotency class two, i.e., its nilpotency class is at most two. Equivalently, the derived subgroup $$[G,G]$$ is contained in the center $$Z(G)$$ of $$G$$.
 * 2) There exists a subgroup $$H$$ of $$G$$ such that $$[G,G] \le H \le Z(G)$$, and every element of $$[G,G]$$ has a unique square root within $$H$$.

Note that for finite groups, this is equivalent to being a LCS-Baer Lie group. However, there are examples of infinite groups that are CS-Baer but not LCS-Baer. The smallest example is central product of UT(3,Z) and Z identifying center with 2Z.