The Hurwitz Theorem

A composition algebra $\KK$ possesses a norm, that is a nondegenerate quadratic form satisfying the identity 1) \begin{equation} |pq|^2 = |p|^2|q|^2 \end{equation} for all $p,q\in\KK$. The Hurwitz Theorem, published posthumously by Adolf Hurwitz in 1923, states that the reals, complexes, quaternions, and octonions are the only real composition algebras with positive-definite norm, and hence the only composition algebras without zero divisors. That is, the only such algebras that contain $\RR$ are $\KK=\RR,\CC,\HH,\OO$.

More generally, all real composition algebras can be obtained from the Cayley-Dickson process — and must have dimension 1,2,4 or 8. Composition algebras of dimension 1 or 2 are both commutative and associative, composition algebras of dimension 4 are associative but not commutative, and composition algebras of dimension 8 are neither. The proof of the Hurwitz Theorem amounts to showing that the Cayley-Dickson process can only yield a composition algebra starting from an associative algebra.

The Hurwitz theorem does however leave open the possibility of composition algebras other than $\RR,\CC,\HH,\OO$, so long as the norm is not positive-definite (or negative-definite, which amounts to the same thing). Furthermore, all such algebras can be constructed using the Cayley-Dixon process, making suitable choices of $\epsilon$ at each step. Starting with the reals, which are 1-dimensional, we can apply the Cayley-Dixon process up to 3 times. Remarkably, however, only 3 distinct new composition algebras are obtained, the so-called split versions of the composition algebras.

1) More correctly, the quadratic form acting on an element $q\in\KK$ gives the squared norm $|q|^2$ of $q$, which can however be positive, negative, or zero; the norm $|q|$ of $q$ makes sense only in the positive-definite case.

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