$S^2 \times S^2$ diffeomorphic to oriented $2$ dimensional vector subspaces of $\mathbb{R}^2$?

Question

As the question title says, is the product of spheres $S^2 \times S^2$ diffeomorphic to the set of oriented $2$ dimensional vector subspaces of $\mathbb{R}^2$?

Answer 1

$S^2\times S^2$ is the Grassmannian of $2$-planes in $R^4$, that is the set of oriented two planes of $R^4$ not of $R^2$.

See a proof here $S^2 \times S^2$ is diffeomophic to $G_2(\mathbb{R}^4)$