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

Statement
Suppose $$L$$ is a Lie ring and $$n$$ is an integer. The map $$x \mapsto nx$$ is an endomorphism of $$L$$ as a Lie ring if and only if the derived subring of $$L$$ has exponent dividing $$n(n-1)$$.

Related facts

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

Analogues in groups

 * nth power map is endomorphism iff abelian (if order is relatively prime to n(n-1))
 * square map is endomorphism iff abelian
 * cube map is automorphism implies abelian
 * cube map is endomorphism iff abelian (if order is not a multiple of 3)
 * inverse map is automorphism iff abelian