Prime power order not implies nilpotent for Lie rings

Statement

There can exist a Lie ring whose order is a power of a prime, but it is not a nilpotent Lie ring.

Related facts

• Prime power order not implies solvable for Lie rings

Proof

Let $p$ be a prime. Consider a Lie ring whose additive group is the elementary abelian group of order $p^{2}$. (alternatively, it is a two-dimensional vector space over the field of $p$ elements), where, for a suitable basis $x,y$, the Lie bracket satisfies:

$[x,y]=x$.

This satisfies all the conditions for a Lie bracket, but it is not nilpotent, because the iterate Lie bracket $[[[\dots [x,y]\dots ],y]$ equals $x$ and never becomes zero.