Bi-Engel Lie ring

Definition

For a pair of values

Suppose $i,j$ are nonnegative integers. An $(i,j)$-bi-Engel Lie ring is a Lie ring $L$ with the property that for any elements $u,x,y\in L$, we have:

$[(\operatorname {ad} u)^{i}x,(\operatorname {ad} u)^{j}y]=0$

where $\operatorname {ad}$ denotes the adjoint action.

Note that being an $(i,j)$-bi-Engel Lie ring is equivalent to being a $(j,i)$-bi-Engel Lie ring.

For a single number

Suppose $n$ is a nonnegative integer. Then:

• A $n$-bi-Engel Lie ring is a Lie ring $L$ that is an $(i,j)$-bi-Engel Lie ring for all pairs of nonnegative integers $i,j$ with $i+j\geq n$.
• A weak $n$-bi-Engel Lie ring is a Lie ring $L$ that is an $(i,j)$-bi-Engel Lie ring for all pairs of positive integers $i,j$ with $i+j\geq n$.

A bounded bi-Engel Lie ring is a Lie ring that is $n$-bi-Engel for some natural number $n$. The smallest $n$ that works is the bi-Engel degree of the Lie ring.

Particular cases

For a pair of numbers

Because $(i,j)$-bi-Engel equals $(j,i)$-bi-Engel, we only list the $i\geq j$ cases.

$i$	$j$	$i+j$	Universal identity for $u,x,y\in L$ for a $(i,j)$-bi-Engel Lie ring	What does it mean to be a $(i,j)$-bi-Engel Lie ring?	What is the largest 3-local nilpotency class that guarantees $(i,j)$-bi-Engel (should be $i+j+1$)?	What is the smallest 3-local nilpotency class guaranteed by $(i,j)$-bi-Engel?	What is the smallest global nilpotency class guaranteed by $(i,j)$-bi-Engel?
0	0	0	$[x,y]=0$	abelian Lie ring	1	1	1
1	0	1	$[[u,x],y]=0$	Lie ring of nilpotency class two	2	2	2
2	0	2	$[[u,[u,x]],y]=0$	same as a (1,2)-Engel-type Lie ring	3	4	4
1	1	2	$[[u,x],[u,y]]=0$	(1,1)-bi-Engel Lie ring	3	need not be 3-locally nilpotent	need not be nilpotent

For a single number

$n$	Universal identities given by bi-Engel conditions for all $u,x,y\in L$	What does it mean to be a $n$-bi-Engel Lie ring?	What is the largest 3-local nilpotency class that guarantees being $n$-bi-Engel? (equal to $n+1$)	What is the smallest 3-local nilpotency class guaranteed by being $n$-bi-Engel?	What is the smallest global nilpotency class guaranteed by being $n$-bi-Engel?
0	$[x,y]=0$	Precisely the same as being an abelian Lie ring	1	1	1
1	$[[u,x],y]=[x,[u,y]]=0$	Precisely the same as being a Lie ring of nilpotency class two	2	2	2
2	$[[u,x],[u,y]]=[[u,[u,x]],y]=[x,[u,[u,y]]]=0$	Precisely the same as being a Lie ring of nilpotency class three	3	3	4