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 $S^5$.