Nice! here is a link: bbc.co.uk/programmes/b00srz5b

Dear BCnrd, I know some theory of semisimple alg. groups (e.g. from Jim's book). But, I dont see how SO_n "magically" reduces to SL_2.

The book of Grove (chapter 7) provides a proof of simplicity only of "real" orthogonal groups PSO_n(R). I don't see the complex case in that book.

In many places they assume "isotropicity" of the group (i.e. Witt index > 0) to get simplicity of the projectivisation of the derived subgroup (L. Grove, thm 6.31, page 58). In the case of usual SO_n, the Witt index is 0, so the simplicity is not clear.

I know this simplicity criterion (due to Tit's) with BN pair buissness, but the main point is the perfectness of the group. I don't know how to show that SO_n's are perfect.