Upper central series members are local powering-invariant in Lie ring

Statement
Let $$L$$ be a Lie ring. All members of the upper central series of $$L$$ are local powering-invariant subrings of $$L$$, i.e., their additive groups are local powering-invariant subgroups of the additive group of $$L$$.

Related facts

 * Second center not is local powering-invariant
 * Upper central series members are local powering-invariant in nilpotent group

Facts used

 * 1) uses::Center is local powering-invariant
 * 2) uses::Local powering-invariant and normal iff quotient-local powering-invariant in nilpotent group
 * 3) uses::Quotient-local powering-invariance is quotient-transitive

Proof
The proof follows directly from Facts (1)-(3).