Is a well known fact that $SO(3)$ acts transitively on $S^2$ and that the isotropy group of this action is $SO(2).$ In this case, $S^2$ has a natural structure of homogeneous space. In particular, I wonder that is true that $SO(3)\times SO(3)$ acts transitively on $S^2\times S^2$ and it has a natural structure of homogeneous space. If we consider in this space the natural $SO(3)-$invariant metric in $S^2$ then we have in $S^2\times S^2$ the product metric and in this metric there are several directions where the curvature is zero.

My question is the following, does there exists a homogeneous structure on $S^2\times S^2$ that is not similar to the product structure? I.e. such that $S^2\times S^2$ is not the quotient of a product of groups?

The point of this questions is that if it is the case, we can search for $G-$invariant metrics oh $S^2\times S^2$ using the possible reductive decomposition of the Lie algebra $\mathfrak{g}$, so it is a manner to search for eventual positively curved metrics.