Ideal property is not transitive for Lie rings

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 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)