Proposition 1

For an A-algebra $F$, the following conditions are equivalent: (a) For every ordered pair of elements $x, y$ of $F$, the subalgebra generated by $x$ and $y$ is associative. (b) The trilinear mapping $(x, y, z) \mapsto a(x, y, z)$ is alternating (§ 7, no. 3). (c) For every ordered pair of elements $x, y$ of $F$, $x^2y = x(xy)$ and $yx^2 = (yx)x$.

Clearly (a) implies (c). We show that (c) implies (b): by definition (§ 7, no. 3) to prove (b), it suffices to verify that $a(x, x, y) = 0$ and $a(x, y, y) = 0$, which is precisely (c).

To prove that (b) implies (a), we use the following 4 lemmas: