Derivation-invariant not implies characteristic

Definition
There can exist a Lie ring $$L$$ with a subring $$S$$ such that $$S$$ is a derivation-invariant Lie subring of $$L$$, such that $$S$$ is not a characteristic subring of $$L$$.

Similar facts

 * Perfect direct factor implies derivation-invariant
 * Self-centralizing direct factor implies derivation-invariant

Converse

 * Characteristic not implies derivation-invariant

Facts used

 * 1) uses::Center is derivation-invariant

Proof
Suppose $$A$$ is a non-abelian Lie ring, $$A_1, A_2$$ are isomorphic copies of $$A$$, and $$L$$ is the direct sum $$A_1 \oplus A_2$$. Define $$S = A_1 +Z(L)$$. Then, $$S$$ is derivation-invariant but not characteristic.

Proof that the subring is not characteristic
Consider the coordinate exchange automorphism that interchanges $$A_1$$ and $$A_2$$. Under this automorphism, $$A_1 + Z(L)$$ goes to $$A_2 + Z(L)$$. Since $$A_2$$ is non-abelian, it is not contained in $$Z(L)$$, so the image of $$A_1 + Z(L)$$ is not equal to it.

Proof that the subring is derivation-invariant
Consider a derivation $$d:L \to L$$. There exist four abelian group endomorphisms $$d_{11}, d_{12}, d_{21}, d_{22}$$ that describe $$d$$, namely:

$$d(x,0) = (d_{11}(x), d_{12}(x)), \qquad d(0,y) = (d_{21}(y),d_{22}(y))$$.

In other words:

$$\! d(x,y) = (d_{11}(x) + d_{21}(y), d_{12}(x) + d_{22}(y))$$.

The derivation condition states that:

$$\! d[(x,y),(x',y')] = [d(x,y),(x',y')] + [(x,y),d(x',y')]$$.

This gives:

$$\! (d_{11}([x,x']) + d_{21}([y,y']), d_{12}([x,x']) + d_{22}([y,y'])) = ([d_{11}(x),x'] + [d_{21}(y),x']+ [x,d_{11}(x')] + [x,d_{12}(y')], [d_{12}(x),y'] + [d_{22}(y),y']) + [y,d_{21}(x')] + [y,d_{22}(y')])$$.

We thus get:

$$\! d_{11}([x,x']) + d_{21}([y,y']) = [d_{11}(x),x'] + [d_{21}(y),x']+ [x,d_{11}(x')] + [x,d_{12}(y')]$$

and:

$$\! d_{12}([x,x']) + d_{22}([y,y']) = [d_{12}(x),y'] + [d_{22}(y),y']) + [y,d_{21}(x')] + [y,d_{22}(y')]$$.

Setting $$y = y' = 0$$ gives that $$d_{11}$$ is a derivation. Setting $$x = x' = 0$$ gives that $$d_{22}$$ is a derivation. Plugging these back in, we get:

$$\! d_{21}([y,y']) = [d_{21}(y),x']+ [x,d_{12}(y')]$$

and:

$$\! d_{12}([x,x']) = [d_{12}(x),y'] + [y,d_{21}(x')]$$.

Setting $$y' = 0$$ in the first equation gives that $$[d_{21}(y),x'] = 0$$ for all $$x',y$$, implying that $$d_{21}$$ takes values in the center of $$A_1$$. Similarly, setting $$x' = 0$$ in the second equation gives that $$d_{12}(x)$$ is in the center of $$A_2$$. In particular, this implies that:

$$\! d(x,0) = (d_{11}(x), d_{12}(x))$$

takes values in $$A_1 \oplus Z(A_2) = A_1 + Z(L) = S$$.

Thus, $$d(A_1) \subseteq S$$. Since $$Z(L)$$ is derivation-invariant by fact (1), $$d(Z(L)) \subseteq Z(L)$$, so $$d(S) = d(A_1) + d(Z(L)) \subseteq S + Z(L) = S$$. Thus, $$S$$ is a derivation-invariant subring of $$L$$.