Center is derivation-invariant

Verbal statement
The center of a Lie ring is a derivation-invariant subring.

Statement with symbols
Let $$L$$ be a Lie ring and $$Z$$ be its center:

$$Z = \{ x \in L \mid [x,y] = 0 \ \forall \ y \in L \}$$.

Then, if $$d:L \to L$$ is a derivation, $$d(Z) \subseteq Z$$.

Related facts

 * Center is invariant under any derivation with partial divided Leibniz condition powers

Proof
Given: A Lie ring $$L$$, its center $$Z = \{ x \in L \mid [x,y] = 0 \ \forall \ y \in L \}$$, a derivation $$d: L \to L$$.

To prove: If $$x \in Z$$, $$dx \in Z$$.

Proof: Pick any $$y \in L$$. Then, we have:

$$d[x,y] = [dx,y] + [x,dy]$$.

Now, since $$x \in Z$$, $$[x,y] = [x,dy] = 0$$. Thus, $$d[x,y] = [x,dy] = 0$$, yielding $$[dx,y] = 0$$.

Thus, $$[dx,y] = 0 \ \forall \ y \in L$$, so $$dx \in Z$$.