LUCS-Baer correspondence

Definition
The LUCS-Baer correspondence is a generalization of the Baer correspondence described as follows:

groups of nilpotency class at two two where every element of the derived subgroup has a unique square root in the group (also known as LUCS-Baer Lie groups) $$\leftrightarrow$$ Lie rings of nilpotency class two where every element of the derived subring has a unique half in the Lie ring (also known as LUCS-Baer Lie rings)