Lie ring of nilpotency class two

Symbol-free definition
A group is said to be of nilpotency class two or nilpotence class two if it satisfies the following equivalent conditions:


 * 1) Its defining ingredient::nilpotency class is at most two, i.e., it is nilpotent of class at most two.
 * 2) Its defining ingredient::derived subring (i.e., the subring generated by all elements arising from the image of the Lie bracket mapping) is contained in its defining ingredient::center
 * 3) The commutator of any two elements of the group is central
 * 4) Any triple commutator (i.e., a commutator where one of the terms is itself a commutator) gives the zero element
 * 5) Its Lie ring of defining ingredient::inner derivations is abelian.