Ideal property is not transitive for Lie rings

This article gives the statement, and possibly proof, of a Lie subring property (i.e., ideal of a Lie ring) not satisfying a Lie subring metaproperty (i.e., transitive Lie subring property).
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ANALOGY: This is an analogue in Lie rings of a fact encountered in group. The old fact is: normality is not transitive.
View other analogues of normality is not transitive|View other analogues from group to Lie ring (OR, View as a tabulated list)

Statement

It is possible to have a Lie ring $L$, an ideal $I$ of $L$, and an ideal $J$ of $I$, such that $J$ is not an ideal of $L$.

Related facts

Related facts about Lie rings

• Derivation-invariant subring of ideal implies ideal
• Left transiter of ideal is derivation-invariant subring
• Derivation-invariance is transitive

Analogues in other algebraic structures

• Normality is not transitive (for groups)