# Invariance under any set of derivations is centralizer-closed

## Statement

### 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 $d \in D$.

## 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.