Normality is not upper join-closed for algebra loops

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 fact about::alternative loop, and in fact, is a fact about::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