Characteristic not implies derivation-invariant

Statement
A characteristic subring of a Lie ring need not be a  derivation-invariant Lie subring.

Similar facts

 * Derivation-invariant not implies characteristic
 * Characteristic not implies ideal

Opposite facts

 * Fully invariant subgroup of additive group of Lie ring is derivation-invariant and fully invariant
 * Characteristic subgroup of additive group of odd-order Lie ring is derivation-invariant and fully invariant
 * Derivation equals endomorphism for Lie ring iff it is abelian
 * Inner derivation implies endomorphism for class two Lie ring

Facts used

 * 1) uses::Characteristic not implies fully invariant in finite abelian group

Proof
By fact (1), there exists a finite abelian group $$G$$ with a characteristic subgroup $$H$$ that is not fully invariant in $$G$$. Consider $$G$$ as an abelian Lie ring, with trivial Lie bracket. Then, the automorphisms of $$G$$ as a Lie ring are the same as the automorphisms as a group, so $$H$$ is a characteristic subring of $$G$$. Further, the derivations of $$G$$ as a Lie ring are precisely its endomorphisms as a group (because the Leibniz rule condition is vacuous). Thus, since $$H$$ is not fully invariant by assumption, $$H$$ is not derivation-invariant.