Concrete description of an exceptional minuscule variety

Question

Let $G$ be a complete reductive Lie group. A simple root $\alpha$ is said to be minuscule if the multiplicity of the coroot $\alpha^\vee$ in $\beta^\vee$ is at most $1$ for all positive roots $\beta$. Associated to a minuscule root $\alpha$ is a maximal parabolic subgroup $P_\alpha$ of $G$, and the quotient $X = G/P_\alpha$ is called a minuscule variety and has various nice properties (see the textbook of Billey and Lakshmibai for details).

In the case $G= SL_n$, all the simple roots are minuscule and the corresponding minuscule varieties are complex Grassmannians, which can be thought of as parameter spaces for $k$-dimensional linear subspaces of an $n$-dimensional complex vector space.

In the other classical types, we can also understand the minuscule varieties $X$ that arise as some sort of parameter space, where for example the points of $X$ correspond to subspaces that are isotropic with respect to some bilinear form.

That leaves exactly two minuscule varieties of exceptional type. One is the projective plane over the octonions, which is not too bad. The other comes from Lie type $E_7$ and I don't know how to interpret it as a parameter space. What is this space? Is there a way to describe it (even non-rigorously) that avoids Lie theory?

Sometimes the name Freudenthal is attached to this space, though I don't know that he studied it explicitly. I have also seen it given the notation $G_\omega(\mathbb{O}^3, \mathbb{O}^6)$, which is suggestive; unfortunately, I've never seen this notation explained anywhere, so I don't know how to interpret it.

Answer 1

One description is $\smash{X=E_7\left/E_6\times S^1\right.}$ (quotient of compact groups, of real dimension 133 – 79 = 54), as removal of the root $\alpha$ in question leaves an $E_6$ diagram. Another is, by Borel-Weil-Tits, $X= G$-orbit of the highest weight line in $\smash{\mathbf{CP}^{55}}$, projective space of the fundamental representation attached to $\alpha$. This is a projective variety whose complex dimension (27), degree (13110), and equations are already in Cartan (1894, p. 144; 1932, p. 160): see Hirzebruch (1957, p. 100).

But the “parameter space description” you want is probably the one given by Tits (1955, p. 138):

Generally if $\beta$ is another simple root, Tits knows that mapping each $x\in G/P_\alpha$ to the “$\alpha$-plane” $\{y\in G/P_\beta: y $ meets $x$ (as cosets inside $G$)$\}$ bijects $G/P_\alpha$ with the set of all $\alpha$-planes in $G/P_\beta$. That makes $X$ a parameter space, and becomes “concrete” if we take for $\beta$ the highest root (at the other end of the $E_7$ diagram): then, he shows, $G/P_\beta$ can be viewed as the “antihermitian quadric” $$ \left\{(x_0,x_1,x_2,y_0,y_1,y_2)\in\mathbf O^6:\sum_{i=0}^2x_i\bar y_i - y_i\bar x_i=0\right\}, $$ projectivized and complexified, and its set of $\alpha$-planes as the set of suitably defined “$\mathbf o$-planes” $\mathbf{OP}^2$ in it. (See p. 89, where these models of $G/P_\beta$ and $G/P_\alpha$ are denoted $\tilde{\mathrm Q a\mathrm h}^5_{\mathbf o}$ and $\tilde{\varGamma a\mathrm h}^{5;2}_{\mathbf o}$.)

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Edit 1:
As you suggest, Freudenthal (1954) already had essentially the same models of $G/P_\alpha$ and $G/P_\beta$ (denoted $\mathfrak M$ and $\mathfrak N$). For a recent exposition see e.g. Omoda (2000, §2.3), where $G$ is denoted $\smash{G^{[2]}}$. Earlier, Freudenthal (1953) also corrected Cartan’s equations of $X$ in $\smash{\mathbf{CP}^{55}}$.

Edit 2:
$X$ also occurs as the special case $\smash{LGr_{\mathbf{CO}}(3)}$ in the “realization of all compact symmetric spaces as ((Double) Lagrangian) Grassmannians” by Huang-Leung (2011) or Eschenburg-Hosseini (2013). Formally this consists of subspaces $\smash{(\mathbf C\otimes\mathbf O)^3\subset(\mathbf H\otimes\mathbf O)^3}$, or the similar thing projectively, but as they discuss, this is subtle to make sense of, as non-associativity makes $\smash{\mathbf O^n}$ not an $\mathbf O$-module.