Complex-subalgebra decomposition in the Albert algebra

Statement

Let $B\subset H_3(\mathbb O)$ be a Jordan subalgebra isomorphic to $H_3(\mathbb C)$ , and suppose that $B$ contains the standard Jordan frame $(e_1,e_2,e_3)$ . Relative to the standard frame decomposition there are real two-dimensional subspaces $V_{12},V_{23},V_{31}\subset\mathbb O$ such that

B=\bigoplus_{i=1}^3\mathbb R e_i \oplus\xi_{12}(V_{12}) \oplus\xi_{23}(V_{23}) \oplus\xi_{31}(V_{31}).

The Jordan-product closure of $B$ forces the cyclic compatibility relations

V_{12}V_{23}\subseteq V_{31}^*,\qquad V_{23}V_{31}\subseteq V_{12}^*,\qquad V_{31}V_{12}\subseteq V_{23}^*,

with the precise placement of conjugations depending on the convention for the maps $\xi_{ij}$ .

A useful normal form

The $\mathrm{Spin}(8)$ subgroup fixing the frame acts on the three off-diagonal copies of $\mathbb O$ through its vector and two half-spin representations. After applying such an automorphism, one may arrange $V_{12}=\mathbb C\subset\mathbb O$ . Choosing a unit vector $a\in V_{23}$ then gives

B=\bigoplus_{i=1}^3\mathbb R e_i \oplus\xi_{12}(\mathbb C) \oplus\xi_{23}(\mathbb C a) \oplus\xi_{31}(a^*\mathbb C).

This is an intermediate normal form: a further frame-fixing automorphism can take $a$ to $1$ , turning $B$ into the standard $H_3(\mathbb C)$ .

Why the decomposition is useful

It converts an embedded Jordan algebra into linear data inside the octonions. The frame decomposition supplies the six summands, while triality coordinates their transformation laws. This is the key input to the transitivity theorem for complex-qutrit subalgebras.

References

John C. Baez and Paul Schwahn, “The Standard Model Gauge Group from the Exceptional Jordan Algebra,” 2026, proof of Lemma 6. arXiv:2606.15235.
Tonny A. Springer and Ferdinand D. Veldkamp, Octonions, Jordan Algebras and Exceptional Groups, Springer, 2000, Chapters 5–7. Publisher record.