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

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

Suppose is a Lie ring with Lie bracket . We say that is the **Lie ring whose bracket is the double of a Lie bracket giving nilpotency class two** if there exists a Lie bracket with the same set and the same additive group structure, such that **both** these conditions are satisfied:

- With the Lie bracket , is a Lie ring of nilpotency class two (i.e., its nilpotency class is at most two).

Note that this *implies* that , with the original Lie bracket, is also of nilpotency class at most two.

## Relation with other properties

### Stronger properties

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

abelian Lie ring | Lie bracket is trivial | (take the half to be the trivial Lie bracket too) |
CS-Baer Lie ring, LCS-Baer Lie ring|FULL LIST, MORE INFO | |

Baer Lie ring | Lie ring of class at most two that is also uniquely 2-divisible, i.e., every element has a unique half |
define | CS-Baer Lie ring, LCS-Baer Lie ring|FULL LIST, MORE INFO | |

LCS-Baer Lie ring | Lie ring of class at most two whose derived subring is uniquely 2-divisible | CS-Baer Lie ring|FULL LIST, MORE INFO | ||

CS-Baer Lie ring | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
|FULL LIST, MORE INFO |

### Weaker properties

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

Lie ring of nilpotency class two | Lie ring arising as the double of a class two Lie cring, Lie ring arising as the skew of a class two near-Lie cring|FULL LIST, MORE INFO | |||

Lie ring whose bracket is the double of a Lie bracket |