LUCS-Baer Lie ring

Definition
A Lie ring $$L$$ is termed a LUCS-Baer Lie ring if it is a Lie ring of nilpotency class two (i.e., the derived subring is contained in the center) and:


 * 1) Every element of its derived subring has a unique half in the whole Lie ring.
 * 2) Every element of its derived subring has a unique half among the elements in the center of the whole Lie ring.
 * 3) Every element of its derived subring has a unique half among the elements in the Lie ring and that unique half is in the center.