All Bianchi type spacetimes have metrics that admits a 3-dimensional killing algebra. They are in general not isotropic. Bianchi type IX have a killing algebra that is isomorphic to SO(3), i.e. the rotation group. But what does it mean? If the fourdimensional spacetime is invariant under the rotation group SO(3) doesn't that imply it is isotropic? My guess is that SO(3) acts as the translation group on our three spatial coordinates, and so the orbits lives in three dimensional space spanned by the spatial coordinates. Therefore, since spacetime includes one extra time coordinate, our three dimensional space is a 3-sphere in four dimensions and SO(3) will be the rotation group on this sphere in four dimensions. But that doesn't seems right, how can SO(3) be the rotation group in both three- and four-dimensional space?