Derivation-invariance is not upper join-closed

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., upper join-closed 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: characteristicity is not upper join-closed.
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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

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