Invariance under any set of derivations is centralizer-closed

For a single derivation
Suppose $$L$$ is a Lie ring, $$S$$ is a subring of a Lie ring, and $$d$$ is a derivation of $$L$$ such that $$d(S) \subseteq S$$. Let $$C = C_L(S)$$ be the centralizer of $$S$$. Then, $$C$$ is also invariant under $$d$$.

For a bunch of derivations
Suppose $$L$$ is a Lie ring, $$S$$ is a subring of a Lie ring, and $$D$$ is a set of derivations of $$L$$ such that $$d(S) \subseteq S$$ for all $$d \in D$$. Let $$C = C_L(S)$$ be the centralizer of $$S$$. Then, $$C$$ is also invariant under all $$d \in D$$.

Particular cases

 * Ideal property is centralizer-closed
 * Derivation-invariance is centralizer-closed

Other related facts

 * Centralizer of a Lie subring is a Lie subring

Analogues in group theory

 * Auto-invariance implies centralizer-closed
 * Normality is centralizer-closed
 * Characteristicity is centralizer-closed

Proof
Note that the statements for a single derivatoin and for a bunch of derivations are clearly equivalent, so we only prove the former.

Given: A Lie ring $$L$$, a subring $$S$$ of $$L$$. $$C = C_L(S)$$ is the set of $$a \in L$$ such that $$[a,s] = 0$$ for all $$s \in S$$. A derivation $$d$$ of $$L$$ such that $$d(S) \subseteq S$$.

To prove: $$d(C) \subseteq C$$.

Proof: 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 $$d(S) \subseteq S$$, $$ds \in S$$, so $$[c,ds] = 0$$. This gives $$[dc,s] = 0$$ as required.