After Bott periodicity is proved, one still has to compute the stable values. For the unitary group $U$, this is easy since you can get away with just $\pi\_0$ and $\pi\_1$. However, I'm having trouble doing this for the orthogonal group. It's easy to work out $\pi\_i(O)$ for, say, $0 \leq i \leq 2$, but already for $i=3$ it takes some work. Can anyone either give me a reference or proof for the computation of $\pi\_i(O)$ for $0 \leq i \leq 7$?



The periodic values in Bott periodicity
Bott's result is best formulated by saying that the sequence of spaces 
$$
\mathbb Z\times BO,\quad O,\quad O/U,\quad U/Sp,\quad \mathbb Z\times BSp,\quad Sp,\quad Sp/U,\quad U/O
$$
are related by the property that each one is the loop space of the previous one (mod 8).


From that, you get for example that $\pi\_4(O)=\pi\_3(O/U)=\pi\_2(U/Sp)=\pi\_1(Z\times BSp)=\pi\_0(Sp)=0$.


