Conjugated cross product on C³

Definition

For $u,v\in\mathbb C^3$ , their conjugated cross product is

u\mathbin{\overline\times}v=\overline{u\times v},

where $u\times v$ is the usual coordinate cross product extended complex-bilinearly and the bar denotes componentwise complex conjugation. Equivalently,

(u\mathbin{\overline\times}v)_i =\sum_{j,k=1}^{3}\varepsilon_{ijk}\,\overline{u_j}\,\overline{v_k}.

Linearity and orthogonality

The operation is real-bilinear and conjugate-linear in each argument over $\mathbb C$ :

(\lambda u)\mathbin{\overline\times}v =\overline\lambda\,(u\mathbin{\overline\times}v), \qquad u\mathbin{\overline\times}(\lambda v) =\overline\lambda\,(u\mathbin{\overline\times}v).

It is alternating. With the Hermitian inner product convention

\langle u,v\rangle=\sum_{i=1}^{3}\overline{u_i}v_i,

one has

\langle u,u\mathbin{\overline\times}v\rangle =\langle v,u\mathbin{\overline\times}v\rangle=0

and the norm identity

|\langle u,v\rangle|^2 +\lVert u\mathbin{\overline\times}v\rVert^2 =\lVert u\rVert^2\lVert v\rVert^2.

Why the conjugation is present

The ordinary complex-bilinear cross product naturally takes values in the dual representation: for $g\in SL(3,\mathbb C)$ ,

(gu)\times(gv)=g^{-T}(u\times v).

For $g\in SU(3)$ , one has $\overline{g^{-T}}=g$ . Componentwise conjugation therefore turns the output into the defining representation, so

g(u\mathbin{\overline\times}v) =(gu)\mathbin{\overline\times}(gv) \qquad(g\in SU(3)).

This equivariance is precisely what makes the $\mathbb C\oplus\mathbb C^3$ octonion product compatible with the standard $SU(3)$ -action.

Convention warning

The bar is part of the operation, not decoration. Omitting it produces a different map and breaks the displayed $SU(3)$ -equivariance in the defining representation. The inner product convention is also important: here it is conjugate-linear in the first argument and linear in the second.

References

John C. Baez and Paul Schwahn, “The Standard Model Gauge Group from the Exceptional Jordan Algebra,” 2026. arXiv:2606.15235. Relevant: §2, especially Lemma 3.
John C. Baez, “Octonions and the Standard Model (Part 2),” 2020. The n-Category Café. Relevant: the Hermitian cross product and $\mathbb C\oplus\mathbb C^3$ multiplication convention.