# Lie ring of nilpotency class three

This article defines a Lie ring property: a property that can be evaluated to true/false for any Lie ring.
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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 2-Engel Lie ring|FULL LIST, MORE INFO
Lie ring of nilpotency class two 2-Engel Lie ring|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