Wouldn't help here, but it's a useful fact anyway.and that can then construct a diffeo from RP^3 to SO(3)

As u say! But the followingWell clearly SO(3) is a subset of R^6, so need show the identity map from SO(3) into R^6 is an embedding

can't be done, as SO(3) as vectors, is contained in R^9. We must show three dimensions vanish.Here it seems that one could just add zeros to the vectors in SO(3), but this must be too simple.

We prove the embedding in the following steps:

(i)

Note that all rotations leave a line fixed (Euler's theorem). This is a nice fact, but irrelevant here

(ii) All elements of can be written as , .

Actually, the x's in the previous relation are related, so call this relation .

Apply the standard matrix operations to to get a matrix of the form . Here are 3x1 vectors, with entries to and to respectively.

The operation is a diffeomorphism between SO(3) and . (Ok, this is long, but at least it's not difficult)

(iii) can be embedded in .

For this, we show that the differential of the inclusion map has maximum rank (=6 here) everywhere.

Consider Then for ,

where is the 6x6 unitary matrix. So has matrix for all , and thus has rank 6.