Frame decomposition of Hermitian Jordan algebras

Core idea

Let $\mathbb K$ be a normed real division algebra and let $(e_1,e_2,e_3)$ be the standard Jordan frame of $H_3(\mathbb K)$ . Write $\xi_{ij}(x)$ for the Hermitian matrix whose $ij$ -entry is $x$ , whose $ji$ -entry is $x^*$ , and whose other entries vanish. Then there is an orthogonal direct sum

H_3(\mathbb K) =\bigoplus_{i=1}^3\mathbb R e_i \;\oplus\; \xi_{12}(\mathbb K)\oplus\xi_{23}(\mathbb K)\oplus\xi_{31}(\mathbb K).

For $\mathbb K=\mathbb O$ , this is the underlying vector-space decomposition of the Albert algebra.

Relation to Peirce spaces

If $(i,j,k)$ is a permutation of $(1,2,3)$ , then

\mathbb R e_i=J_1(e_i), \qquad \xi_{ij}(\mathbb K)=J_{1/2}(e_i)\cap J_{1/2}(e_j),

and

J_0(e_k) =\mathbb R e_i\oplus\mathbb R e_j\oplus\xi_{ij}(\mathbb K).

Thus the decomposition simultaneously refines the three Peirce decompositions.

Multiplication between pair spaces

With a cyclic choice of indices and a compatible convention for the maps $\xi_{ij}$ , matrix multiplication gives

\xi_{ij}(x)\circ\xi_{jk}(y)=\tfrac12\,\xi_{ik}(xy).

Changing which entry receives $x$ rather than $x^*$ changes the displayed conjugations and order, but not the structural point: multiplying adjacent pair spaces lands in the third pair space. In the octonionic case these three eight-dimensional spaces carry the vector and two half-spin representations under the frame-fixing $\mathrm{Spin}(8)$ .

References

John C. Baez and Paul Schwahn, “The Standard Model Gauge Group from the Exceptional Jordan Algebra,” 2026, §4. arXiv:2606.15235.
Tonny A. Springer and Ferdinand D. Veldkamp, Octonions, Jordan Algebras and Exceptional Groups, Springer, 2000, Chapters 5–7. Publisher record.