Showing SO(3) Subset is Projective Plane Diffeomorphic

[Xang]

I was wondering if anyone can help me to show that the subset of SO(3) contaning all
matrices A with det(A+id)=0 is a submanifold diffeomorphic to real projective plane.
Thanks.

 

[Hurkyl]

Well, what ideas have you had on it so far? Where are you stuck?

 

[Xang]

Actually, I could show that SO(3) is homeo. to RP^3. But on don't know why the condition det(A+id)=0 implies that this subset would be diffeo. to RP^2.