Center is local powering-invariant in Lie ring

Statement
Suppose $$L$$ is a Lie ring and $$Z(L)$$ is its center. Then, $$Z(L)$$ is a local powering-invariant subring of $$L$$. Explicitly, if $$a \in Z(L)$$ and $$n$$ is a natural number such the equation $$nx = a$$ has a unique solution for $$x \in L$$, then that unique solution $$x$$ is in $$Z(L)$$.

Related facts

 * Center is local powering-invariant
 * Derived subring is divisibility-invariant in Lie ring