Artin's theorem on alternative algebras

Remarks

The idea of the proof: the subalgebra is spanned by the products of copies of $x$ and $y$ with all bracketings. One shows that the associator of any three such products is $0$, by induction on their total length, using that the associator is alternating and the Moufang identities. The converse also holds: if every subalgebra generated by two elements is associative, then $(x, x, y) = (y, x, x) = 0$, so the algebra is alternative. Example: in the octonions, any two elements lie in an associative subalgebra, which is isomorphic to $\R$, $\C$ or the quaternions. The statement fails for three elements: the octonion units $i$, $j$ and $\ell$ satisfy $(ij)\ell \neq i(j\ell)$.

Going deeper

For a student

Taking $y = x$ shows that every alternative algebra is power-associative. In the octonions a nonzero $x$ has the inverse $x^{-1} = \bar{x}/(x\bar{x})$, which is a combination of $1$ and $x$. So the theorem, applied to $x$ and $y$, gives $x^{-1}(xy) = y = (yx)x^{-1}$. This is the reason why the nonzero octonions behave like a group in many computations, although they only form a Moufang loop. The theorem is a key step in the study of alternative division algebras and composition algebras.