3-Engel and (2,5)-torsion-free implies class six for Lie rings

Statement
There are many versions:

Related facts

 * 2-Engel implies class three for Lie rings
 * 2-Engel and 3-torsion-free implies class two for Lie rings
 * n-Engel and torsion-free threshold at least n implies solvable iff nilpotent for Lie rings

Facts used

 * 1) uses::Higgins' lemma on Engel conditions
 * 2) uses::Higgins' lemma on annihilator of polynomial being an ideal
 * 3) uses::2-Engel implies class three for Lie rings
 * 4) uses::2-Engel implies third member of lower central series is in 3-torsion

Proof
For all these proofs, we will follow the right-normed convention for expressing Lie products, and hence avoid putting unnecessary Lie brackets and parentheses. Note that Higgins' original proof uses the left-normed convention instead, so the ordering of symbols in this proof is the mirror image of that in Higgin's original proof.

Original proof: class at most six
Given: A 3-Engel (2,5)-torsion-free Lie ring $$L$$.

To prove: $$L$$ is a nilpotent Lie ring and its nilpotency class is at most six.

Proof:

For this proof, we will follow the right-normed convention for expressing Lie products, and hence avoid putting unnecessary Lie brackets and parentheses.

Slight improvement: class at most five
Given: A 3-Engel (2,5)-torsion-free Lie ring $$L$$.

To prove: $$L$$ is a nilpotent Lie ring and its nilpotency class is at most five.

Proof:

(2,3,5)-torsion-free implies class four
Given': A 3-Engel (2,3,5)-torsion-free Lie ring $$L$$.

To prove: $$L$$ has nilpotency class at most four.

3-torsion implies class four
Given': A 3-Engel Lie ring such that $$[L,[L,[L,L]]]$$ is in the 3-torsion of $$L$$.

To prove: $$L$$ has nilpotency class at most four.

Proof:

(2,5)-torsion-free Lie algebra over a field implies class four
The proof follows by splitting into two cases:


 * The characteristic of the field is not 3: In this case, we can use the (2,3,5)-torsion-free case
 * The characteristic of the field is 3: In this case, we can use the 3-torsion case

Journal references

 * , Theorem 3
 * , Theorem 2.3