Determinant of a two-by-two octonionic Hermitian matrix

The positive cone of $H_2(\mathbb O)$ is the closure of the cone of matrices with $a>0$ and $\det A>0$. Equivalently, $A$ is positive semidefinite when its associated real quadratic form on $\mathbb O^2$ is nonnegative.

Because the determinant is quadratic, it has a symmetric bilinear polarization

This degree-two determinant and its polarization are the algebraic operations used in the octonionic Monge–Ampère measure.

This determinant belongs to the rank-two spin factor. The Albert algebra $H_3(\mathbb O)$ has a cubic Jordan determinant, but there is no analogous ordinary determinant on arbitrary-size octonionic Hermitian matrices.

1. Semyon Alesker, “Plurisubharmonic functions on the octonionic plane and $\operatorname{Spin}(9)$-invariant valuations on convex sets,” Journal of Geometric Analysis 18 (2008), 651–686. arXiv record. Relevant: §1.2.
2. Jacques Faraut and Adam Korányi, Analysis on Symmetric Cones, Oxford University Press, 1994. Relevant: Euclidean Jordan algebras and spin factors.