In Relativity and Singularities, Natário states that
Any vacuum solution admitting $O(n)$ as an isometry group is locally isometric to $M=\mathbb{R}^2\times S^{n-1}$ (with the Schwarzchild metric).
Why is that so? It doesn't seem obvious to me, and, so far, all "proofs" on the web approach this problem from the physical point of view, so the connection to the above statement is blurred to me.