# Lie ring arising as the double of a class two Lie cring

From Groupprops

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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

## Definition

A **Lie ring arising as the double of a class two Lie cring** is defined as a Lie ring such that there exists the structure of a class two Lie cring with the same additive group and underlying set , such that the Lie bracket of is the double of the cring operation, i.e., if the cring operation is , then:

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

abelian Lie ring | ||||

Baer Lie ring | ||||

LCS-Baer Lie ring | ||||

Lie ring whose bracket is the double of a Lie bracket giving nilpotency class two |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Lie ring arising as the skew of a class two near-Lie cring | ||||

Lie ring of nilpotency class two | double of class two Lie cring is class two Lie ring |