Hurwitz's theorem on normed division algebras

Statement

Hurwitz's theorem. Every finite-dimensional unital real algebra with a positive-definite multiplicative quadratic norm is isomorphic, as a normed real algebra, to exactly one of

\mathbb R,\qquad \mathbb C,\qquad \mathbb H,\qquad \mathbb O.

Consequently, a real normed division algebra has dimension $1$ , $2$ , $4$ , or $8$ .

What the theorem does not assume

Associativity and commutativity are not hypotheses. The conclusion explains their progressive failure: $\mathbb R$ and $\mathbb C$ are commutative and associative, $\mathbb H$ is associative but noncommutative, and the octonions are noncommutative and nonassociative but alternative.

Composition-algebra form

Over a field of characteristic different from $2$ , every unital composition algebra has dimension $1$ , $2$ , $4$ , or $8$ . The real theorem above adds positive definiteness, which rules out split forms and makes every nonzero element invertible.

Relation to sums of squares

Choosing an orthonormal basis turns norm multiplicativity into an identity expressing a product of two sums of $n$ squares as another sum of $n$ squares with bilinear entries. Hurwitz's theorem is therefore also called the $1$ - $2$ - $4$ - $8$ theorem for composition of quadratic forms.

References

Adolf Hurwitz, “Über die Composition der quadratischen Formen von beliebig vielen Variabeln,” Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen (1898), 309–316. Digitized record.
John C. Baez, “The Octonions,” Bulletin of the American Mathematical Society 39 (2002), 145–205. DOI record. Relevant: §2.2.