Derivation-invariance does not satisfy intermediate subring condition

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 $$B$$.

Other similar facts

 * Derivation-invariance is not upper join-closed

Analogues in other algebraic structures

 * Characteristicity does not satisfy intermediate subgroup condition