3-locally nilpotent Lie ring

Definition
A Lie ring $$L$$ is termed a 3-locally nilpotent Lie ring if the subring of $$L$$ generated by any subset of $$L$$ of size at most three is a defining ingredient::nilpotent Lie ring.

If there is a common bound on the nilpotency class for all such subrings, then the smallest common bound is termed the 3-local nilpotency class.