Say a manifold M has 3 vector fields S,T and R whose Lie brackets satisfy the equations $[S,T]=R$, $[R,S]=T$ and $[T,R]=S$

Then I suppose the following properties hold for M,

There exists a metric on M whose Killing Fields are $S$,$T$ and $R$

There exists a foliation of M with manifolds on which $SO(3)$ has a transitive action.

There are many possible loose ends in the above statements, like the metric can be pseudo-Riemannian (surely Schwarzschild Metric is an example which satisfies the above) and if M is a $4$-manifold then the foliation is probably only by 2-spheres.

I guess this is an application/special case of the Frobenius Theorem or its dual.

I would like to know what is the precise statement along these lines and its proof (reference) and if there is some general framework in which this fits in. (like for some arbitrary group instead of just $SO(3)$)

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