are going to compute.

Auxiliary computations
----------------------

For any $n\in\N$, let $S^n=\R[x_0,\ldots,x_n]/(\sum x_i^2-1)$. Using the long exact sequence of localization, we see that it suffices to compute the Witt groups of $S^{n-1}$ and $S^n_{x_n}$ to compute the Witt groups of $S^n$. The ring $S^n_{x_n}$ has a nice expression:

If $B^n=\R[x_0,\ldots,x_n]/(\sum_{i=0}^{n-2} x_i^2-x_{n-1}x_n+1)$ (for $n\geq 1$), there is an isomorphism $$\psi:S^n_{x_n}\to B^n_{x_{n-1}+x_n}$$ defined by $$\psi(x_i)=\left\{ \begin{array}{ll} \frac {2x_i}{(x_{n-1}+x_n)} & \mbox{for any $0\leq i\leq n-2$,}\\
 \frac {x_n-x_{n-1}}{(x_{n-1}+x_n)} & \mbox{if $i=n-1$,}\\
 \frac 2{(x_{n-1}+x_n)} & \mbox{if $i=n$.}
 \end{array}\right.$$ Its inverse is the homomorphism $\phi:B^n_{x_{n-1}+x_n}\to S^n_{x_n}$ defined by $$\phi(x_i)=\left\{ \begin{array}{cl} \frac {x_i}{x_n} & \mbox{for any $0\leq i\leq n-2$,}\\
 \frac {1-x_{n-1}}{x_n} & \mbox{if $i=n-1$,}\\
 \frac {1+x_{n-1}}{x_n} & \mbox{if $i=n$.}
 \end{array}\right.$$ The computation of $H^i(B^n,W)$ with $n$ small is not very difficult. To achieve this, we need an additional auxiliary ring $C^n=\R[x_0,\ldots,x_n]/(\sum x_i^2+1)$.

\[comput\_c1\] We have $$H^i(C^1,W)=\left\{ \begin{array}{cl} \Z/4 & \mbox{if $i=0$,}\\
0 & \mbox{if $i\geq 1$.}\end{array}\right.$$

Since $C^1$ is a curve, $H^i(C^1,W)=0$ for $i\geq 2$. We also get $H^0(C^1,W)=W^0(C^1)=\Z/4$, generated by the class of $\langle 1\rangle$ by [@Mon Theorem 2.9]. To compute $H^1$, we first observe that $H^1(C^1,I)=H^1(C^1,W)$ by Lemma \[deg\_1\].

Using [@EL Theorem 2’], we get that $I^2(\kappa(C^1))=0$. Moreover, since $C^1$ has no real point then 