Lemma 4

Suppose that the mapping $a$ is alternating. Then, for all $x \in \mathbf{F}$, the subalgebra of $\mathbf{F}$ generated by $x$ is strongly associative.

$a(u, v, w) = 0$ whenever two of the three elements $u, v, w$ are equal to $x$ and it suffices to apply Lemma 3.