I am attempting to read Atiyah's paper on self-duality in four-dimensional Riemannian geometry, and I came across the following basic example:

Let $S_-$ be the $SU(2)$-bundle of anti-self dual spinors over $S^4$. Then the total space of the projectivised bundle $PS_{-}$ over $S^4$ is $\mathbb{C}P^3$.

I was wondering if someone could give a brief proof/explanation of this fact (after searching the internet I find many references to this fact, but no complete explanations)? I understand that the quaternionic Hopf fibration exhibits $\mathbb{C}P^3$ as an $S^2$-bundle over $S^4$, but I only vaguely see that this bundle is in fact the projectivised ASD spinor bundle. (Basically I have tried to write down transition functions for both for the usual splitting of $S^4 = \mathbb{H}P^1$ into two hemispheres. I was hoping that someone could give a better reason and/or the details of this procedure.)

Thanks in advance, and sorry if this question is basic! Everyone seems to just claim that the total space is obviously $\mathbb{C}P^3$, but to me it is not obvious :(.

(In a similar vein, if someone could describe and explain the twistor space of $\mathbb{C}P^2$, that would also be great!)