I'm not well versed in projective geometry since it is not really my field. I read in [1] about the existance of a projective plane $\left(\mathbb{C}\otimes\mathbb{O}\right)P^{2}$ defined on bi-quaternions. I think it might be defined through the use of the Jordan algebra $\mathfrak{h}_{3}\left(\mathbb{C}\otimes\mathbb{O}\right)$ but I'm wondering if there's some reference for a direct construction of such "projective plane" or if its existance is merely speculative.

[1] Baez, J. C. .2002. The octonions. Bull. Amer. Math. Soc. 39.

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    $\begingroup$ It is the compact Riemannian symmetric space associated with the real form of $E_6$ whose maximal compact subgroup is (some finite quotient of?) $U_1 × \mathit{Spin}_{10}$, see this question. I'm not sure there is a known construction from the algebra $\mathbb{C} \otimes \mathbb{C}$, however. $\endgroup$
    – Gro-Tsen
    Jun 24, 2021 at 14:37
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    $\begingroup$ (In the tables of this Wikipedia article, it is known as “EIII”.) $\endgroup$
    – Gro-Tsen
    Jun 24, 2021 at 14:39
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    $\begingroup$ It's also known as the "first Rosenfeld plane", see arxiv.org/abs/2007.05451 for information about its homotopy type and some references. $\endgroup$
    – skupers
    Jun 24, 2021 at 14:41


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