How many Killing spinors exist on $S^5$?

Question

So, I know that on $S^n$, a spinor of the form

$$ \Sigma^\pm = \frac{1 \pm i\gamma^\alpha z_\alpha}{\sqrt{1+z^2}}\Sigma_0$$

where $\Sigma_0$ is a constant spinor, is a Killing spinor on $S^n$ because it satisfies

$$D_a \Sigma^\pm = \pm i \gamma_a \Sigma^\pm$$

where $D_a$ is the covariant derivative (with the spin connection). But how many such Killing spinors exist?

I think the $\pm$ signs yield two linearly independent solutions. But for any sign choice, there ought to be $2^{D}$ spinor components in $SO(2D)$ and $SO(2D+1)$. I am confused by the fact that the sphere is really a coset manifold: $$S^n = SO(n+1)/SO(n).$$

How does one connect spinors of $SO(n+1)$ and $SO(n)$ to those of $$S^n = SO(n+1)/SO(n)~?$$

This post imported from StackExchange Physics at 2015-03-10 12:58 (UTC), posted by SE-user leastaction