Derivation-invariance is centralizer-closed

This article gives the statement, and possibly proof, of a Lie subring property (i.e., derivation-invariant subring) satisfying a Lie subring metaproperty (i.e., centralizer-closed Lie subring property)
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ANALOGY: This is an analogue in Lie rings of a fact encountered in group. The old fact is: characteristicity is centralizer-closed.
Another analogue to the same fact, in the same new context, is: characteristicity is centralizer-closed for Lie rings
View other analogues of characteristicity is centralizer-closed|View other analogues from group to Lie ring (OR, View as a tabulated list)

Statement

Verbal statement

The centralizer of a derivation-invariant subring of a Lie ring is also derivation-invariant.

Related facts

• Ideal property is centralizer-closed

Weaker facts

• Center is derivation-invariant

Proof

Given: A Lie ring $L$, a derivation-invariant subring $S$ of $L$. $C=C_{L}(S)$ is the set of $a\in L$ such that $[a,s]=0$ for all $s\in S$.

To prove: $C$ is also derivation-invariant, i.e., $d(C)\subseteq C$ for any $d\in \operatorname {Der} (L)$.

Proof: Suppose $d\in \operatorname {Der} (L)$. For any $c\in C$ and $s\in S$, we need to show that $[dc,s]=0$.

For this, note that, by the Leibniz rule property of derivations:

$d([c,s])=[dc,s]+[c,ds]$.

Since $[c,s]=0$, the left side is zero. Further, since $S$ is derivation-invariant, $ds\in S$, so $[c,ds]=0$. This gives $[dc,s]=0$ as required.