Lemma 2

Let $E$ be an A-algebra such that the trilinear mapping $(x, y, z) \mapsto a(x, y, z)$ is alternating, $S$ a generating system of $E$ and $U$ a sub-A-module of $E$ containing $S$ and such that $sU \subset U$ and $Us \subset U$ for all $s \in S$. Then $U = E$.

The set $U'$ of $x \in E$ such that $xU \subset U$ and $Ux \subset U$ is obviously a sub-A-module of $E$, which contains $S$ by hypothesis. On the other hand, for $x, y$ in $U'$ and $u \in U$, by hypothesis

on passing to the opposite algebra, we have similarly $u(xy) \in U$. Hence $U'$ is a subalgebra of $E$ and, since it contains $S$, $U' = E$. Hence $EU \subset U$ and a fortiori $UU \subset U$, which proves that $U$ is a subalgebra of $E$; as it contains $S$, $U = E$, which proves the lemma.

A subset $H$ of $F$ is called strongly associative if $a(u, v, w) = 0$ when at least two of the elements $u, v, w$ belong to $H$.