Octonionic spin factor

Definition

The octonionic spin factor is the ten-dimensional real Jordan algebra

\mathfrak h_2(\mathbb O) =\left\{ \begin{pmatrix}\alpha&z\\ \bar z&\beta\end{pmatrix} :\alpha,\beta\in\mathbb R,\ z\in\mathbb O \right\}, \qquad X\circ Y=\frac12(XY+YX).

It is a Euclidean Jordan algebra despite the nonassociativity of the octonions.

Spin-factor identification

Set $\lambda=(\alpha+\beta)/2$ and

u=((\alpha-\beta)/2,z)\in\mathbb R\oplus\mathbb O\cong\mathbb R^9.

The matrix corresponds to $(\lambda,u)$ , and its Jordan product becomes the spin-factor product. Consequently

\mathfrak h_2(\mathbb O)\cong J(\mathbb R^9).

It is simple, has rank $2$ , dimension $10$ , and abstract automorphism group $O(9)$ .

Embedding in the exceptional algebra

The upper-left corner gives a Jordan embedding

\mathfrak h_2(\mathbb O)\hookrightarrow\mathfrak h_3(\mathbb O), \qquad X\longmapsto \begin{pmatrix}X&0\\0&0\end{pmatrix}.

It is not unital: the source unit becomes $\operatorname{diag}(1,1,0)$ , a proper idempotent of the Albert algebra.

The setwise stabilizer of this corner inside $\operatorname{Aut}(\mathfrak h_3(\mathbb O))\cong F_4$ is $\operatorname{Spin}(9)$ . This ambient stabilizer is not the abstract automorphism group $O(9)$ of the spin factor: its action on the whole 27-dimensional Albert algebra contains additional information.

Degree caveat

Alternativity is enough for the degree-two product to satisfy the Jordan identity. Degree three also works and gives the Albert algebra, but the same Hermitian-matrix construction fails to be Jordan in degrees at least four.

References

Tonny A. Springer and Ferdinand D. Veldkamp, Octonions, Jordan Algebras and Exceptional Groups, Springer, 2000. Publisher record.
John C. Baez and Paul Schwahn, “The Standard Model Gauge Group from the Exceptional Jordan Algebra,” 2026. arXiv:2606.15235.