Normality is not upper join-closed for algebra loops

ANALOGY BREAKDOWN: This is the breakdown of the analogue in algebra loops of a fact encountered in group. The old fact is: normality is upper join-closed.
View other analogue breakdowns of normality is upper join-closed|View other analogue breakdowns from group to algebra loop

Statement

It is possible to have the following: an algebra loop $L$, a subloop $N$ of $L$, two subloops $M_{1},M_{2}$ of $L$ containing $N$, such that $N$ is a normal subloop in both $M_{1}$ and $M_{2}$ but $N$ is not a normal subloop of the subloop generated by $M_{1}$ and $M_{2}$.

In fact, we can construct a counterexample $L$ that is an Alternative loop (?), and in fact, is a Moufang loop (?).

Related facts

• Normality is upper join-closed
• Normality satisfies intermediate subloop condition
• Ideal property is upper join-closed for Lie rings
• Ideal property is not upper join-closed for alternating rings

Proof

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