Normality is not upper join-closed for algebra loops

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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

Proof

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