Multiplication by n map is a derivation iff derived subring has exponent dividing n

Statement
Suppose $$L$$ is a Lie ring and $$n$$ is an integer. The multiplication by $$n$$ map, i.e., the map $$x \mapsto nx$$, is a derivation of $$L$$ if and only if the additive group of the derived subring of $$L$$ has exponent dividing $$n$$.

Related facts

 * Multiplication by n map is an endomorphism iff derived subring has exponent dividing n(n-1)