Lie ring of nilpotency class three

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Definition

A Lie ring of nilpotency class three is a Lie ring $L$ satisfying the following equivalent conditions:

1. Its nilpotency class is at most three. This is equivalent to checking the identity: $\![w,[x,[y,z]]]=0\ \forall \ w,x,y,z\in L$
2. Its 3-local nilpotency class is at most three. In other words, the subring generated by any subset of size at most three is a nilpotent Lie ring of nilpotency class at most three.
3. The following identities hold for all $x,y,z\in L$:
   • $[x,[y,[y,z]]]=0$
   • $[x,[y,[x,z]]]=0$
4. The Lie ring is a 2-bi-Engel Lie ring, i.e., the following hold for all $u,x,y\in L$:
   • $[[u,[u,x]],y]=0$
   • $[[u,x],[u,y]]=0$
   • $[x,[u,[u,y]]]=0$

Equivalence of definitions

Further information: Nilpotency class three is 3-local for Lie rings

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
abelian Lie ring				|FULL LIST, MORE INFO
Lie ring of nilpotency class two				|FULL LIST, MORE INFO
2-Engel Lie ring		2-Engel implies class three for Lie rings		|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Lie ring of 2-local nilpotency class three
metabelian Lie ring
(1,1)-bi-Engel Lie ring
3-Engel Lie ring