It is proved in Weinberg's text (Gravitation and cosmology, ch.13 section 5) that if a manifold has a submanifold which is maximally symmetric (has maximal number of killing vector fields w.r.t. the metric $g$ restricted to the submanifolds i.e., $g_2$ ) then it is essentially a warped product, the metric can be written as
$$g = g_1 \oplus fg_2.$$
However, the proof provided therein is really long and involves solving explicit PDEs. And I have not seen even a mention of this anywhere else yet this seems to be very helpful in constraining the metric from the assumed symmetry from the outset. Is there a better proof anywhere?
It seems to me that the Killing condition $L_X g=0$ should be able to give me all of the constraints, but I cannot discern anything about the full metric other than $g_1$ part wouldn't depend on coordinates of the submanifold. Can this result be proven using this or the Killing equation?