Jordan–von Neumann–Wigner classification

Remarks

Proof idea: a finite-dimensional formally real Jordan algebra has a unit and a positive definite trace form, so ideals have orthogonal complements that are again ideals. This gives the decomposition into simple ideals. A simple algebra is then studied through a complete system of orthogonal primitive idempotents and the Peirce decomposition it induces. With two idempotents one gets a spin factor. With at least three idempotents the off-diagonal spaces carry a composition algebra, which is $\R$, $\C$, $\mathbb{H}$ or $\mathbb{O}$ by Hurwitz's theorem. The octonions occur only with exactly three idempotents, because they are not associative.

Example: the Hermitian $2 \times 2$ complex matrices form a simple formally real Jordan algebra of dimension $4$. In the list it appears as the spin factor $\R \oplus \R^3$, with the Pauli matrices as an orthonormal basis of $\R^3$. Similarly $H_2(\R)$, $H_2(\C)$ and $H_2(\mathbb{H})$ are the spin factors with $n = 2, 3, 5$, which is why item 3 starts at $n = 3$.

Going deeper

For a student

The theorem is the Jordan analogue of the classification of compact simple Lie groups, and the two are linked. The automorphism groups of the simple pieces are compact Lie groups, for example $F_4$ for the Albert algebra. Through the Koecher–Vinberg theorem, the list also classifies the irreducible symmetric cones, such as the cone of positive definite matrices and the Lorentz cone. The same list of simple pieces classifies the simple Euclidean Jordan algebras, since these are the same as the simple formally real ones.