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.