The Stiefel manifolds are presented in this Wikipedia article over the division algebras $\mathbb{R,C,H}$. In fact, they are presented as homogeneous spaces, respectively for the $A,B,C$,and $D$ series Lie groups. Do analogous results exist for the octonions and the exceptional Lie groups?

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    $\begingroup$ I don't understand the close vote - the question is clear enough. Unfortunately, the only related spaces I know are $\mathbb{OP}^2=F_4/\mathrm{Spin}(9)$ and maybe $\mathbb{OP}^1$. Their construction is entirely different, so it is at least not obvious how to cook up a Stiefel manifold from them. $\endgroup$ Sep 5, 2019 at 20:01
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    $\begingroup$ One way that this question could be 'improved' would be for the OP to provide a list of the properties that Stiefel manifolds have that are desired in the exceptional cases. For example, the Stiefel manifolds $V_k({\mathbb F})$ for ${\mathbb F}={\mathbb R},{\mathbb C}, {\mathbb H}$ all have the property that $V_0({\mathbb F})$ is a point and there is a submersion of $G$-homogeneous spaces $V_{k+1}({\mathbb F})\to V_k({\mathbb F})$ with fibers that are spheres, when $G$ is one of the classical groups. Such a sequence exists for ${\mathrm G}_2$, but not for ${\mathrm F}_4$ or ${\mathrm E}_k$. $\endgroup$ Sep 16, 2019 at 11:17


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