Octonions as complex vectors

Core idea

As a real vector space, the octonions can be written

\mathbb O\cong\mathbb C\oplus\mathbb C^3.

Using the Hermitian product $\langle u,v\rangle=\sum_i \overline{u_i}v_i$ , conjugate-linear in its first argument, and the conjugated cross product $u\mathbin{\overline\times}v=\overline{u\times v}$ , define multiplication by

(a,u)(b,v) = \left( ab-\langle u,v\rangle, \overline a\,v+b u+u\mathbin{\overline\times}v \right),

where all scalar-vector products on the right are the ordinary complex scalar action. This is the convention used in the cited construction; in particular, conjugation falls on the scalar $a$ from the left factor.

Why this is an octonion algebra

The identity is $(1,0)$ , conjugation is

(a,u)^*=(\overline a,-u),

and the squared norm is

\lVert(a,u)\rVert^2=|a|^2+\langle u,u\rangle.

The cross-product norm identity gives

\lVert(a,u)(b,v)\rVert =\lVert(a,u)\rVert\,\lVert(b,v)\rVert.

This makes $\mathbb C\oplus\mathbb C^3$ an eight-dimensional real normed division algebra, so Hurwitz's theorem identifies it with $\mathbb O$ .

Geometric origin of the splitting

Choose a unit imaginary octonion $i$ . Then $\operatorname{span}_{\mathbb R}\{1,i\}$ is an embedded copy of $\mathbb C$ , and its orthogonal complement is six-dimensional. In the displayed convention, right multiplication by $i$ equips that complement with the complex structure corresponding to ordinary multiplication by $i$ on $\mathbb C^3$ ; left multiplication corresponds to multiplication by $-i$ . The resulting decomposition is orthogonal for the octonion inner product.

Convention warning

This is a direct sum of real vector spaces and the product is real-bilinear, not complex-bilinear. In particular, the second component contains $\overline a\,v$ , not $a v$ , and contains the conjugate $\overline{u\times v}$ , not the ordinary cross product. Moving the conjugate-linear slot of the Hermitian inner product or choosing the opposite complex structure changes several terms at once; such variants can still describe an isomorphic octonion algebra but must not be mixed term by term.

Symmetry

The standard action $g(a,u)=(a,gu)$ of $SU(3)$ preserves every operation in the product, hence acts by octonion automorphisms and fixes the $\mathbb C$ summand pointwise. This realizes the $SU(3)$ stabilizer of a chosen complex octonion subalgebra.

References

John C. Baez and Paul Schwahn, “The Standard Model Gauge Group from the Exceptional Jordan Algebra,” 2026. arXiv:2606.15235. Relevant: §2 and Lemma 3.
John C. Baez, “Octonions and the Standard Model (Part 2),” 2020. The n-Category Café. Relevant: the explicit $\mathbb C\oplus\mathbb C^3$ multiplication convention.
Ichiro Yokota, Exceptional Lie Groups, Lecture Notes in Mathematics 2369, Springer, 2025. arXiv:0902.0431. Relevant: Theorem 1.9.1 and the structures underlying the construction.