the real algebraic sphere of dimension $3$ satisfy the hypothesis of the theorem. We first need some preparation in order to compute the group $\K {S^3}$. This is the object of the next section.

The Gersten-Witt complex {#gwss}
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The complex
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Let $X$ be an integral regular scheme over a field $k$. Let $d=\dim X$. The *Gersten-Witt complex* is the complex $$\xymatrix@C=1.3em{0\ar[r] & W(\kappa(X))\ar[r] & \displaystyle{\bigoplus_{x_1\in X^{(1)}} W(\kappa(x_1))   }\ar[r] & \ldots  \ar[r] &  \displaystyle{\bigoplus_{x_d\in X^{(d)}} W(\kappa(x_d))   }\ar[r] & 0}$$ as constructed in [@BW]. Observe that the Witt groups in the complex are in fact twisted Witt groups, i.e. Witt groups with twisted duality $\homm {\kappa(x_p)}{\_}{\omega_{x_p}}$, where $\omega_{x_p}=\exten p{\O_{X,x_p}}{\kappa(x_p)}{\O_{X,x_p}}$ (which is a $\kappa(x_p)$-vector space of dimension $1$ because $X$ is regular). We denote by $H^i(X,W)$ the cohomology groups of this complex. If $d\leq 3$, we have that $H^i(X,W)=W^i(X)$ for $0\leq i\leq 3$ ([@BW] again).

Even if the Witt groups in the complex are twisted by line bundles, we can still define the fundamental ideal and the groups behaving like the powers of it (just choose any isomorphism between the twisted Witt group and the classical Witt group, see for instance [@Fa1 Definition 9.2.1]). It turns out that the differentials of 