CharlesGWaldman on Nostr: To me, it's just a circle bundle over a sphere - the first vector is the image of ...

To me, it's just a circle bundle over a sphere - the first vector is the image of e₁ under the rotation, which lies on a 2-sphere in R³, then the image of e₂ is a unit vector orthogonal to this vector, so we can think of it lying on a unit circle in the tangent plane at that point. (The image of e₃ is determined by the images of e₁ and e₂ by orthonormality and orientation). This has always been the easiest way to visualize SO(3) to me. No identifications required, you just have to be comfortable with bundles...