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The group $Spin(9)$ is a subgroup of $SO(16)$ and acts transitively on the unit sphere $S^{15}$. $Spin(9)$ acts naturally on the space of octonionic Hermitian $2\times 2$-matrices (I do not define this action here).

Does the latter action diagonalize any octonionic Hermitian matrix of size $2\times 2$?

A reference would be most helpful.

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Yes, in fact, any $2$-by-$2$ octonionic Hermitian matrix is equivalent under the natural $\mathrm{Spin}(9)$ action to a diagonal $2$-by-$2$ octonionic Hermitian matrix.

This follows from the well-known fact that $\mathrm{F}_4$, the automorphism group of the Jordan algebra $H_3(\mathbb{O})$, can diagonalize any element of $H_3(\mathbb{O})$, and $\mathrm{Spin}(9)\subset\mathrm{F}_4$ is the stabilizer of an idempotent $e$ of trace $1$.

A good reference is the book by F. Reese Harvey, Spinors and Calibrations. I think his discussion is in the final chapter of the book, but I don't have it with me at the moment.

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  • $\begingroup$ IIRC there are some errors in Harvey regarding around $F_4$.. It is a good introduction though. For more explicit and general calculations I like the monograph by Ichiro Yokota. arxiv.org/abs/0902.0431 $\endgroup$
    – Vít Tuček
    Apr 24, 2023 at 18:22

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