Ideal 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., ideal of a Lie ring) need not satisfy the second Lie subring property (i.e., derivation-invariant Lie subring)
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ANALOGY: This is an analogue in Lie rings of a fact encountered in group. The old fact is: normal not implies characteristic.
View other analogues of normal not implies characteristic|View other analogues from group to Lie ring (OR, View as a tabulated list)

Statement

There can exist a Lie ring $L$ and a subring $I$ of $L$ such that $I$ is an ideal of $L$ and is not a derivation-invariant subring of $L$.

Related facts

Similar facts for Lie rings

• Ideal property is not transitive for Lie rings
• Characteristic not implies derivation-invariant
• Characteristic ideal not implies derivation-invariant

Converse and related facts

• Derivation-invariant subring implies ideal
• Derivation-invariant subring of ideal implies ideal

Related facts for groups and other algebraic structures

• Normal not implies characteristic

Proof

Let $L$ be an abelian Lie ring whose additive group is the Klein four-group. Thus, $L=A\oplus B$ where $A,B$ are subrings of size two. Since $L$ is abelian, both $A$ and $B$ are ideals of $L$.

Let $\sigma$ be the automorphism of $L$ that interchanges $A$ and $B$. Then, $\sigma$ is a derivation of $L$, but $A$ and $B$ are not invariant under $\sigma$.