Determinant of a two-by-two octonionic Hermitian matrix

For an element

A=\begin{pmatrix}a&q\\ \bar q&b\end{pmatrix} \in [[nonassociative-algebra/octonionic-spin-factor|H_2(\mathbb O)]], \qquad a,b\in\mathbb R,\quad q\in\mathbb O,

its determinant is the real quadratic polynomial

\det A=ab-|q|^2.

Positivity

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.

Polarization

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

\det(A,B)=\frac12\bigl(\det(A+B)-\det A-\det B\bigr).

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

Dimension warning

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.

References

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.
Jacques Faraut and Adam Korányi, Analysis on Symmetric Cones, Oxford University Press, 1994. Relevant: Euclidean Jordan algebras and spin factors.