Derived subring is divisibility-invariant in Lie ring

Statement
Suppose $$L$$ is a Lie ring and $$L' = [L,L]$$ is its derived subring. Then, $$L'$$ is a divisibility-invariant subring of $$L$$.

Related facts

 * Derived subgroup is divisibility-invariant in nilpotent group
 * Center is local powering-invariant in Lie ring