# Lie ring of nilpotency class three

From Groupprops

This article defines a Lie ring property: a property that can be evaluated to true/false for any Lie ring.

View a complete list of properties of Lie ringsVIEW RELATED: Lie ring property implications | Lie ring property non-implications |Lie ring metaproperty satisfactions | Lie ring metaproperty dissatisfactions | Lie ring property satisfactions | Lie ring property dissatisfactions

## Contents

## Definition

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

- Its nilpotency class is at most three. This is equivalent to checking the identity:
- 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.
- The following identities hold for all :
- The Lie ring is a 2-bi-Engel Lie ring, i.e., the following hold for all :

### 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 |