Derivation-invariance does not satisfy intermediate subring condition

This article gives the statement, and possibly proof, of a Lie subring property (i.e., derivation-invariant Lie subring) not satisfying a Lie subring metaproperty (i.e., intermediate subring condition).
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ANALOGY: This is an analogue in Lie rings of a fact encountered in group. The old fact is: characteristicity does not satisfy intermediate subgroup condition.
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Statement

There can exist a Lie ring $L$, a derivation-invariant Lie subring $A$ of $L$, and a subring $B$ of $L$ containing $A$, such that $A$ is a derivation-invariant Lie subring of $B$.

Related facts

Other similar facts

• Derivation-invariance is not upper join-closed

Analogues in other algebraic structures

• Characteristicity does not satisfy intermediate subgroup condition