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for completeness and to satisfy links from Calabi-Penrose fibration
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Couldn’t one explain this via $S^7 = Sp(2)/Sp(1)$ (unit sphere in $\mathbb{H}^2\simeq \mathbb{C}^4$) and then $\mathbb{CP}^3 = S^7/U(1)$, using (I assume) the fact $U(1)$ is central? Or do you want the explicit statement as written? One could do a non-compact version, with $GL(2,\mathbb{H})/Sp(1) \stackrel{?}{=} \mathbb{H}^2 \setminus \{0\}\simeq \mathbb{C}^4 \setminus \{0\}$ and then $\mathbb{CP}^3 = \mathbb{C}^4 \setminus \{0\}/\mathbb{C}^\times$ …
Just to clarify that I am not worried about the explanation of this fact but about citing it properly.
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where it’s equation (3)
Try Таблица 1 of
where we have, essentially, $C_n/(C_{n-1}\oplus T)= \mathbf{PC}^{2n-1}$, where $C_n=Sp(n)$ and $T=U(1)$.
I’m sure its much older than this, though.
Oh, well, then this: https://hal.archives-ouvertes.fr/hal-00121742 (See section 3), which appeared as a chapter in the Handbook of Pseudo-Riemannian Geometry and Supersymmetry would be good. More specifically: the discussion before Prop 3.2 proves what you want, I think.
Added Or else see
in particular table 3 (which reproduces the result from the 1960s)
Thanks again!
The pointer to Butruille I already had in the entry, in its arXiv version (but this is one of those reference that just takes it all for granted).
The Onishchik Encyclopedia is good. Now I remember that we we had been citing this table elsewhere already (such as at SU(3)). Thanks for reminding me! Have added it here now, too.
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