I'd posted on that in the thread "Riemannian curvature of maximally symmetric spaces".
One first finds an equation for the second derivative of Killing vector ξ. This means that ξ anywhere is a function of ξ and its first derivative at some point. The defining equation for ξ constrains the first derivative of ξ to be antisymmetric or sort-of antisymmetric where ξ itself is 0. For n space dimensions:
Possible values of ξ: n
Possible values of the antisymmetrized first derivative of ξ: n(n-1)/2