N-local nilpotency class n implies nilpotency class n

Statement

For groups

Suppose $G$ is a group and $n$ is a natural number such that the $n$-local nilpotency class of $G$ is at most $n$. Then, $G$ is a nilpotent group and the nilpotency class of $G$ is at most $n+1$.

For Lie rings

Suppose $L$ is a Lie ring and $n$ is a natural number such that the $n$-local nilpotency class of $L$ is at most $n$. Then, $L$ is a nilpotent Lie ring and the nilpotency class of $L$ is at most $n+1$.

Facts used

1. 2-Engel implies class three for groups, 2-Engel implies class three for Lie rings

Proof

The proof follows mostly from Fact (1).