Continuing Joel Fine and Fran Burstall's answer about, indeed "neat", Lebrun's result. Just want to recall that the "orthogonal" twistor space of any $2n$-dimensional pseudo-sphere $SO(2p+1,2q)/SO(2p,2q)$ can be written as $SO(2p+2,2q)/U(p+1,q)$. So the Kähler manifold in question, in case of the 6-sphere, is $SO(8)/U(4)$. One should think of each $j:T_xS^6\rightarrow T_xS^6$ as a linear map on $R^8$ with $j(x)=-1$ and $j(1)=x$. Well, proofs have been rewritten of LeBrun's result. I wish I had more opinion on this:

>R. Albuquerque, Isabel M.C. Salavessa, _On the twistor space of pseudo-spheres_ Differential Geometry and its Applications, **25** (2007), pp. 207-219, doi:[10.1016/j.difgeo.2006.08.004](https://doi.org/10.1016/j.difgeo.2006.08.004), arXiv:[math/0509442](http://arxiv.org/abs/math/0509442).