Write down the explicit form of the $15$ Killing vectors of the 5-sphere I am looking for a way to write down explicitly the $15$ vectors which are generators of $SO(6)$ in polar coordinates on the $5$-sphere.
In particular I have the round metric $$g_{\mu\nu} = \left(
\begin{array}{ccccc}
 1 & 0 & 0 & 0 & 0 \\
 0 & \sin ^2(\zeta ) & 0 & 0 & 0 \\
 0 & 0 & \cos ^2(\zeta ) & 0 & 0 \\
 0 & 0 & 0 & \cos ^2(\alpha ) \sin ^2(\zeta ) & 0 \\
 0 & 0 & 0 & 0 & \sin ^2(\alpha ) \sin ^2(\zeta ) \\
\end{array}
\right)$$
in coordinates $x^\mu = \left(\zeta, \alpha, \varphi_1,\varphi_2,\varphi_3\right)$. And I know it admits 15 Killing vectors, generators of $SO(6)$, as the sphere can be written as a coset space of groups, i.e. $S^5 \cong \frac{SO(6)}{SO(5)}$.
My problem, now is to find the explicit form of these vectors in these coordinates. I managed to work out the solution for the analogous case of a $3$-sphere, but I am in trouble for $S^5$.
 A: The general procedure for getting Killing vectors is to solve the Killing equations: $$\mathcal{L}_K g = 0$$If you're given a metric, you can try to solve these equations for the unknown vector fields, but it gets messy quickly. A shortcut, which easily finds some Killing vectors, is to note that if a metric is independent of a coordinate $x$, then the corresponding vector field $\partial_x := \frac{\partial}{\partial x}$ is a Killing vector field. In the above example this is enough to find that $\partial_{\phi_1}, \partial_{\phi_2}, \partial_{\phi_3}$ are all Killing vector fields. 
That's 3 Killing vectors, but if you want them all you are going to have to solve those equations. Personally, as soon as anything gets over 2 or 3 dimensions, I resort to using a computer. Using Maple, my computer was able to spit out a list of 15 Killing vectors for the metric you've written down in a few seconds. Maple isn't so great at trig simplification, but you can easily do that by hand.
Here is some of the output from the quick calculation:

