How do the oriented orthonormal frame bundles of product spaces $M=M_{1}\times M_{2}$ work?
For example, I'm interested in the manifold $M=M_{1}\times M_{2}=S^{3}\times S^{1}$. I know that the oriented orthonormal frame bundle of each one $OF(M_{i})$ is:
$$OF(S^{3})=SO(4)$$
where:
$$S^{3}=\frac{SO(4)}{SO(3)}$$
And
$$OF(S^{1})=SO(2)$$
Note, that I'm working with a pseudoriemannian metric such that $S^{1}$ is associated with negative signature in the quadratic form.
I'm guessing, in analogy with Minkowski space $R^{3,1}$ that we have:
$$OF(S^{3}\times S^{1})=SO(4,2)$$
Is it then going to be true that:
$$SU(2)\times U(1)=\frac{SO(4,2)}{SO(3,1)}$$
Where we have used the isomorphisms $SU(2)=S^{3}$ and $S^{1}=U(1)$.
Or, put another way, can we then write the Principle oriented orthonormal frame bundle as:
$$OF(S^{3}\times S^{1})=SU(2)\times U(1)\times SO(3,1)$$
However clearly:
$$SO(4,2)\neq SU(2)\times U(1)\times SO(3,1)$$
??? can someone explain what is the correct way? How would I map this (the OF(M) to the tangent bundle?