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There are two compact exceptional Hermitian symmetric spaces. The complexified octonionic projective plane $\mathrm{E_6/U(1)Spin(10)}$:

Q1: What exactly is the complexified octonionic projective plane?

Q2: Can we describe explicitly the Riemannian metric using octonions? This can be answered in two possible ways:

  1. The isotropy representation can be written down using complexified octonions so how does the $\mathrm{Spin(10)}$-invariant inner product look like?
  2. This space can be identified as a minimal closed orbit in the complex projectivization of the complex exceptional Jordan algebra $\mathcal{J}_3(\mathbb{O_C})$. (This minimal orbit should be given by $[A] \in \mathbb{CP}^{26}$ for $A \in \mathcal{J}_3(\mathbb{O_C})$ of "octonionic rank 1.") Is the Riemannian metric in question pullback of some $\mathrm{E_6}$-invariant object that lives on this complex Jordan algebra? I'd prefer this way as it would extend the case of real octonionic planes that I've considered in my previous question.

and the other one $\mathrm{E_7/U(1)E_6}$:

Q3: Is there a nice model of this space as in the case Q2.2?

Q4: The isotropy representation is the complexified Jordan algebra. What is the $\mathrm{E_6}$-invariant scalar product?

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    $\begingroup$ As far as 2, goes, I assume you mean the projectivization of a minimal closed orbit in the complex exceptional Jordan algebra. In that case, yes, as Élie Cartan explained in his 1894 thesis, if you take the $\mathrm{E}_6$-invariant cubic form $J$ on this algebra and look at the cone cut out by the Jacobian ideal of $J$ (a linear system of quadrics), this is a 17-dimensional cone that projectivizes to the (minimal) 16-dimensional $\mathrm{E}^\mathbb{C}_6$-orbit in $\mathbb{CP}^{26}$. It is homogeneous under any maximal compact subgroup of $\mathrm{E}^\mathbb{C}_6$. $\endgroup$ – Robert Bryant Jun 30 '17 at 19:37
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    $\begingroup$ For the $\mathrm{E}_7$ symmetric space, look at its minimal irreducible representation, which has (complex) dimension $56$. In this representation, the group preserves a quartic form Q (and a symplectic form, but you don't need that). If you let $I$ be the ideal generated by the quadrics that are second partials of $Q$, it cuts out a cone of dimension $28$. When you projectivize it, you get the 27-dimensional homogeneous space of $\mathrm{E}^\mathbb{C}_7$ embedded into $\mathbb{CP}^{55}$. It is homogeneous under any maximal compact in $\mathrm{E}^\mathbb{C}_7$. (Also in Cartan's thesis.) $\endgroup$ – Robert Bryant Jun 30 '17 at 19:45
  • $\begingroup$ @RobertBryant Yes, I forgot about projectivization. Thank you. Now that you've written down the $\mathrm{E_7}$ case I remember seeing this representation under the name Freudenthal vector space. $\endgroup$ – Vít Tuček Jul 1 '17 at 8:13

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