Derivation-invariance is not upper join-closed

Statement
It is possible to have a Lie ring $$L$$, a subring $$I$$ of $$L$$, and subrings $$A,B$$ of $$L$$ such that $$I$$ is a derivation-invariant Lie subring of both $$A$$ and $$B$$, but $$I$$ is not derivation-invariant in the Lie subring generated by $$A$$ and $$B$$.

Related facts for Lie rings

 * Ideal property is upper join-closed for Lie rings
 * Derivation-invariance does not satisfy intermediate subring condition

Analogues in other algebraic structures

 * Characteristicity is not upper join-closed