Lie ring of nilpotency class three

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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