Group theory and Spacetime symmetries

[cesiumfrog]

"A manifold (with a metric tensor) is said to be spherically symmetric iff the Lie algebra of its Killing vector fields has a sub-algebra that is the Lie algebra of SO(3)." Why?

The statement is paraphrased from texts such as Schutz or D'Inverno, where it is always expressed like a definition with no explanation.. except to note that this definition is sufficient to calculate the general form of a spherically symmetric metric/line-element (rather than just writing one down intuitively), which then might be used (in combined with the vacuum-EFE) to obtain Schwarzschild's solution. Could anyone point me to an explanation or proof for the statement?

It does make sense that there should be some relationship between the space-time symmetries and the familiar rotation group, but why do we know the relationship to have this specific form?

 

[shoehorn]

"A manifold (with a metric tensor) is said to be spherically symmetric iff the Lie algebra of its Killing vector fields has a sub-algebra that is the Lie algebra of SO(3)." Why?

Think about the more familiar definition of spherical symmetry that one finds in, say, Wald:

A spacetime is spherically symmetric if its isometry group contains a subgroup isomorphic to SO(3) and whose orbits are 2-spheres.

(At least I think that's how Wald defines it; it's certainly the first definition that trips off my tongue, and I know that I learned this kind of thing from Wald.) Can you see how to relate the statement you qouted to the one I've given here?

Presumably, for example, you're aware that the symmetries of a (pseudo-)Riemannian manifold are encoded in the set of Killing vectors (if it exists) of that manifold. And presumably you know that if a complete set of Killing vectors encodes all the information about the symmetries of the manifold, then a subset of the set of Killing vectors could encode information about a smaller symmetry such as the rotational invariance of the manifold?

Can you see a way to combine this fact with the fact that a natural operation on pairs of vector field is the Lie bracket? And can you see how this will lead you to -- at the very least -- an intuitive acceptance of the statement?