Characteristic not implies derivation-invariant

This article gives the statement and possibly, proof, of a non-implication relation between two Lie subring properties. That is, it states that every Lie subring satisfying the first Lie subring property (i.e., characteristic subring of a Lie ring) need not satisfy the second Lie subring property (i.e., derivation-invariant Lie subring)
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Statement

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

Related facts

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. 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.