SlowGuess/NeurIPS_2023_OCR huggingface dataset:
For  $w \in \Gamma(V, \mathfrak{q})$  we now apply  $\rho(w) \in \mathrm{GL}(V)$  to the basis elements  $e_i$ . Note that by definition of  $\Gamma(V, \mathfrak{q})$  we have  $\rho(w)(e_i) \in V$ . With this we get:
$$
\begin{array}{l} 2 \mathfrak {b} (\rho (w) \left(e _ {i}\right), \rho (w) \left(e _ {j}\right)) = \rho (w) \left(e _ {i}\right) \rho (w) \left(e _ {j}\right) + \rho (w) \left(e _ {j}\right) \rho (w) \left(e _ {i}\right) (473) \\ = \rho (w) \left(e _ {i} e _ {j} + e _ {j} e _ {i}\right) (474) \\ = \rho (w) \left(2 \mathfrak {b} \left(e _ {i}, e _ {j}\right)\right) (475) \\ = 2 \mathfrak {b} \left(e _ {i}, e _ {j}\right). (476) \\ \end{array}
$$
This shows for  $v = \sum_{i=1}^{n} a_i \cdot e_i$ :
$$
\begin{array}{l} \mathfrak {q} (\rho (w) (v)) = \mathfrak {q} \left(\sum_ {i = 1} ^ {n} a _ {i} \cdot \rho (w) \left(e _ {i}\right)\right) (477) \\ = \sum_ {i = 1} ^ {n} a _ {i} ^ {2} \cdot \mathfrak {q} (\rho (w) \left(e _ {i}\right)) (478) \\ = \sum_ {i = 1} ^ {n} a _ {i} ^ {2} \cdot \mathfrak {q} \left(e _ {i}\right) (479) \\ = \mathfrak {q} \left(\sum_ {i = 1} ^ {n} a _ {i} \cdot e _ {i}\right) (480) \\ = \mathfrak {q} (v). (481) \\ \end{array}
$$
Since  $\rho(w)$  is also a linear automorphism of  $V$  with inverse  $\rho(w^{-1})$  we see that:
$$
\rho (w) | _ {V} \in \mathrm {O} (V, \mathfrak {q}). \tag {482}
$$
By Lemma E.23 we also see that:
$$
\rho (w) | _ {\mathcal {R}} = \mathrm {i d} _ {\mathcal {R}}. \tag {483}
$$
Together this shows that:
$$
\rho (w) | _ {V} \in \mathrm {O} _ {\mathcal {R}} (V, \mathfrak {q}). \tag {484}
$$
$$
\operatorname {r a n} (\rho : \Gamma (V, \mathfrak {q}) \rightarrow \mathrm {G L} (V)) \subseteq \mathrm {O} _ {\mathcal {R}} (V, \mathfrak {q}). \tag {485}
$$
Recall the definition of the set of radical preserving orthogonal automorphisms:
$$
\begin{array}{l} \mathrm {O} _ {\mathcal {R}} (V, \mathfrak {q}) := \left\{\Phi \in \mathrm {O} (V, \mathfrak {q}) \mid \Phi | _ {\mathcal {R}} = \mathrm {i d} _ {\mathcal {R}} \right\} \cong \left( \begin{array}{c c} \mathrm {O} (E, \mathfrak {q} | _ {E}) & 0 _ {m \times r} \\ \mathrm {M} (r, m) & \mathrm {i d} _ {\mathcal {R}} \end{array} \right) (486) \\ \cong \mathrm {O} (E, \mathfrak {q} | _ {E}) \ltimes \mathrm {M} (r, m). (487) \\ \end{array}
$$
So an element  $\Phi \in \mathrm{O}_{\mathcal{R}}(V,\mathfrak{q})$  can equivalently be written as:
$$
\begin{array}{l} \Phi = \left( \begin{array}{c c} O & 0 \\ M & I \end{array} \right) = \left( \begin{array}{c c} O & 0 \\ 0 & I \end{array} \right) \circ \left( \begin{array}{c c} I & 0 \\ M & I \end{array} \right) (488) \\ = \left( \begin{array}{c c} O _ {1} & 0 \\ 0 & I \end{array} \right) \circ \dots \circ \left( \begin{array}{c c} O _ {k} & 0 \\ 0 & I \end{array} \right) \circ \left( \begin{array}{c c} I & 0 \\ M _ {1, 1} & I \end{array} \right) \circ \dots \circ \left( \begin{array}{c c} I & 0 \\ M _ {m, r} & I \end{array} \right), (489) \\ \end{array}
$$
where  $O = O_{1} \cdots O_{k}$  is a product of  $k \leq m$  reflection matrices  $O_{l}$  by the Theorem of Cartan-Dieudonne C.13, and  $M = \sum_{i=1}^{m} \sum_{j=1}^{r} M_{i,j}$  where the matrix  $M_{i,j} = c_{i,j} \cdot I_{i,j}$  only has the entry  $c_{i,j} \in \mathbb{F}$  at  $(i,j)$  (and 0 otherwise). Now let  $w_{l} \in V$  be the normal vector of the reflection  $O_{l}$  with  $\mathfrak{q}(w_{l}) \neq 0$  for  $l = 1, \ldots, k$ , and for  $i = 1, \ldots, m$  and  $j = 1, \ldots, r$  put:
$$
\gamma_ {i, j} := 1 + c _ {i, j} \cdot e _ {i} e _ {m + j} \in \operatorname {C l} (V, \mathfrak {q}), \tag {490}
$$
and further:
$$
w := w _ {1} \dots w _ {k} \cdot \gamma_ {1, 1} \dots \gamma_ {m, r} \in \operatorname {C l} (V, \mathfrak {q}). \tag {491}
$$
Note that by Examples E.19 and E.20 we have:
$$
\{w \in V \mid \mathfrak {q} (w) \neq 0 \} \subseteq \Gamma (V, \mathfrak {q}), \tag {492}
$$
$$
\left\{\gamma_ {i} := 1 + e _ {i} \sum_ {j = 1} ^ {r} c _ {i, j} e _ {m + j} \mid c _ {i, j} \in \mathbb {F}, i = 1, \dots , m, j = 1, \dots , r \right\} \subseteq \Gamma (V, \mathfrak {q}), \tag {493}
$$
which implies that  $w\in \Gamma (V,\mathfrak{q})$  .With this we get:
$$
\begin{array}{l} \rho (w) = \rho \left(w _ {1}\right) \circ \dots \circ \rho \left(w _ {k}\right) \circ \rho \left(\gamma_ {1, 1}\right) \circ \dots \circ \rho \left(\gamma_ {m, r}\right) (494) \\ = \left( \begin{array}{c c} O _ {1} & 0 \\ 0 & I \end{array} \right) \circ \dots \circ \left( \begin{array}{c c} O _ {k} & 0 \\ 0 & I \end{array} \right) \circ \left( \begin{array}{c c} I & 0 \\ M _ {1, 1} & I \end{array} \right) \circ \dots \circ \left( \begin{array}{c c} I & 0 \\ M _ {m, r} & I \end{array} \right) (495) \\ = \Phi . (496) \\ \end{array}
$$
This thus shows the surjectivity of the map:
$$
\rho : \Gamma (V, \mathfrak {q}) \rightarrow \mathrm {O} _ {\mathcal {R}} (V, \mathfrak {q}). \tag {497}
$$
We can summarize our finding in the following statement.
Corollary E.26. We have the short exact sequence:
$$
1 \longrightarrow \bigwedge^ {[ \times ]} (\mathcal {R}) \xrightarrow {\text {i n c l}} \Gamma (V, \mathfrak {q}) \xrightarrow {\rho} \mathrm {O} _ {\mathcal {R}} (V, \mathfrak {q}) \longrightarrow 1. \tag {498}
$$
From the above Theorem we can now also derive the structure of the elements of the Clifford group. Corollary E.27 (Elements of the Clifford group). Let  $(V, \mathfrak{q})$  be a finite dimensional quadratic vector space of dimension  $n := \dim V < \infty$  over a fields  $\mathbb{F}$  with  $\mathrm{char}(\mathbb{F}) \neq 2$ . Let  $\mathcal{R} \subseteq V$  be the radical vector subspace of  $(V, \mathfrak{q})$  with dimension  $r := \dim \mathcal{R}$ . Put  $m := n - r \geq 0$ . Let  $e_1, \ldots, e_n$  be an orthogonal basis of  $(V, \mathfrak{q})$  ordered in such a way that  $e_{m+1}, \ldots, e_{m+r} = e_n$  are the basis vectors inside  $\mathcal{R}$ , while  $e_1, \ldots, e_m$  are spanning a non-degenerate orthogonal subspace to  $\mathcal{R}$  inside  $(V, \mathfrak{q})$ .
Then every element of the Clifford group  $w \in \Gamma(V, \mathfrak{q})$  is of the form:
$$
w = c \cdot v _ {1} \dots v _ {k} \cdot \gamma_ {1} \dots \gamma_ {m} \cdot g, \tag {499}
$$
with  $c\in \mathbb{F}^{\times}$ $k\in \mathbb{N}_0$ $v_{l}\in V$  with  $\mathfrak{q}(v_l)\neq 0$  for  $l = 1,\ldots ,k$
$$
\gamma_ {i} = 1 + e _ {i} \sum_ {j = 1} ^ {r} c _ {i, j} e _ {m + j}, \tag {500}
$$
with  $c_{i,j}\in \mathbb{F}$  for  $i = 1,\ldots ,m$ $j = 1,\dots ,r$  , and some  $g\in \bigwedge^{[*]}(\mathcal{R})$
E.4 Orthogonal Representations of the Clifford Group
Lemma E.28. Let  $e_1, \ldots, e_n$  be an orthogonal basis of  $(V, \mathfrak{q})$  and  $w \in \Gamma(V, \mathfrak{q})$ . If we put for  $j \in [n]$ :
$$
b _ {j} := \rho (w) \left(e _ {j}\right), \tag {501}
$$
and for  $A\subseteq [n]$
$$
b _ {A} := \prod_ {i \in A} ^ {<  } b _ {i} = \rho (w) \left(e _ {A}\right), \tag {502}
$$
then  $b_{1}, \ldots, b_{n}$  is an orthogonal basis for  $(V, \mathfrak{q})$  and both  $(e_{A})_{A \in [n]}$  and  $(b_{A})_{A \in [n]}$  are orthogonal bases for  $(\operatorname{Cl}(V, \mathfrak{q}), \bar{\mathfrak{q}})$ .
Proof. First note that  $b_{1}, \ldots, b_{n}$  is a basis of  $V$ . Indeed, the relation:
$$
0 = \sum_ {i = 1} ^ {n} a _ {i} \cdot b _ {i}, \tag {503}
$$
with  $a_{i}\in \mathbb{F}$  implies:
$$
\begin{array}{l} 0 = \rho (w) ^ {- 1} (0) (504) \\ = \rho (w) ^ {- 1} \left(\sum_ {i = 1} ^ {n} a _ {i} \cdot b _ {i}\right) (505) \\ = \sum_ {i = 1} ^ {n} a _ {i} \cdot \rho (w) ^ {- 1} \left(b _ {i}\right) (506) \\ = \sum_ {i = 1} ^ {n} a _ {i} \cdot e _ {i}. (507) \\ \end{array}
$$
Since  $e_1, \ldots, e_n$  is linear independent we get  $a_i = 0$  for all  $i \in [n]$ . So also  $b_1, \ldots, b_n$  is linear independent and thus constitute a basis of  $V$ .
To show that  $b_{1}, \ldots, b_{n}$  is an orthogonal basis of  $(V, \mathfrak{q})$  let  $i \neq j$  and then consider the following:
$$
\begin{array}{l} 2 \cdot \mathfrak {b} \left(b _ {i}, b _ {j}\right) = b _ {i} b _ {j} + b _ {j} b _ {i} (508) \\ = \rho (w) \left(e _ {i}\right) \rho (w) \left(e _ {j}\right) + \rho (w) \left(e _ {j}\right) \rho (w) \left(e _ {i}\right) (509) \\ = \rho (w) \left(e _ {i} e _ {j} + e _ {j} e _ {i}\right) (510) \\ = \rho (w) \left(2 \cdot \underbrace {\mathfrak {b} \left(e _ {i} , e _ {j}\right)} _ {= 0}\right) (511) \\ = 0. (512) \\ \end{array}
$$
This shows that  $b_{1}, \ldots, b_{n}$  is an orthogonal basis of  $(V, \mathfrak{q})$ .
Theorem D.26 then shows that both  $(e_A)_{A\in [n]}$  and  $(b_{A})_{A\in [n]}$  are orthogonal bases for  $(\operatorname {Cl}(V,\mathfrak{q}),\bar{\mathfrak{q}})$ .
Theorem E.29. Let  $w \in \Gamma(V, \mathfrak{q})$  and  $x \in \operatorname{Cl}(V, \mathfrak{q})$  then we get:
$$
\bar {\mathfrak {q}} (\rho (w) (x)) = \bar {\mathfrak {q}} (x). \tag {513}
$$
In other words,  $\rho (w)\in \mathrm{O}(\mathrm{Cl}(V,\mathfrak{q}),\bar{\mathfrak{q}})$ . Furthermore, we have:
$$
\rho (w) | _ {\Lambda (\mathcal {R})} = \operatorname {i d} _ {\Lambda (\mathcal {R})}. \tag {514}
$$
In other words,  $\rho (w)\in \mathrm{O}_{\Lambda (\mathcal{R})}(\mathrm{Cl}(V,\mathfrak{q}),\bar{\mathfrak{q}})$
Proof. Let  $e_1, \ldots, e_n$  be an orthogonal basis for  $(V, \mathfrak{q})$  and  $b_j := \rho(w)(e_j)$  for  $j \in [n]$ . Then by Lemma E.28 we know that  $b_1, \ldots, b_n$  is an orthogonal basis of  $(V, \mathfrak{q})$  and both  $(e_A)_{A \subseteq [n]}$  and  $(b_A)_{A \subseteq [n]}$  are orthogonal basis for  $(\operatorname{Cl}(V, \mathfrak{q}), \bar{\mathfrak{q}})$ . Now let  $x \in \operatorname{Cl}(V, \mathfrak{q})$  and write it as:
$$
x = \sum_ {A \subseteq [ n ]} x _ {A} \cdot e _ {A}, \quad \rho (w) (x) = \sum_ {A \subseteq [ n ]} x _ {A} \cdot \rho (w) \left(e _ {A}\right) = \sum_ {A \subseteq [ n ]} x _ {A} \cdot b _ {A}. \tag {515}
$$
Then we get:
$$
\begin{array}{l} \bar {\mathfrak {q}} (\rho (w) (x)) = \sum_ {A \subseteq [ n ]} x _ {A} \cdot \prod_ {i \in A} \mathfrak {q} \left(b _ {i}\right) (516) \\ = \sum_ {A \subseteq [ n ]} x _ {A} \cdot \prod_ {i \in A} \mathfrak {q} (\rho (w) \left(e _ {i}\right)) (517) \\ = \sum_ {A \subseteq [ n ]} x _ {A} \cdot \prod_ {i \in A} \rho (w) \left(e _ {i}\right) \rho (w) \left(e _ {i}\right) (518) \\ = \sum_ {A \subseteq [ n ]} x _ {A} \cdot \prod_ {i \in A} \rho (w) \left(e _ {i} ^ {2}\right) (519) \\ = \sum_ {A \subseteq [ n ]} x _ {A} \cdot \prod_ {i \in A} \rho (w) (\mathfrak {q} \left(e _ {i}\right) \cdot 1) (520) \\ = \sum_ {A \subseteq [ n ]} x _ {A} \cdot \prod_ {i \in A} \mathfrak {q} \left(e _ {i}\right) \cdot \rho (w) (1) (521) \\ = \sum_ {A \subseteq [ n ]} x _ {A} \cdot \prod_ {i \in A} \mathfrak {q} \left(e _ {i}\right) (522) \\ = \bar {\mathbf {q}} (x). (523) \\ \end{array}
$$
This shows the claim. The remaining point follows from Lemma E.23.
Similarly, and more detailed, we also get the following, using Theorem D.27, Corollary E.17 and Lemma E.23:
Corollary E.30. For every  $w \in \Gamma(V, \mathfrak{q})$  and  $m = 0, \ldots, n$  we have:
$$
\rho (w) | _ {\operatorname {C l} ^ {(m)} (V, \mathfrak {q})} \in \mathrm {O} _ {\bigwedge^ {(m)} (\mathcal {R})} \left(\operatorname {C l} ^ {(m)} (V, \mathfrak {q}), \bar {\mathfrak {q}}\right). \tag {524}
$$
In words,  $\rho(w)$ , when restricted to the  $m$ -th homogeneous multivector component  $\mathrm{Cl}^{(m)}(V,\mathfrak{q})$  of  $\mathrm{Cl}(V,\mathfrak{q})$  acts as an orthogonal automorphism of  $\mathrm{Cl}^{(m)}(V,\mathfrak{q})$  w.r.t.  $\bar{\mathfrak{q}}$ . Furthermore, it acts as the identity when further restricted to the  $m$ -th homogeneous multivector component of the radical subalgebra:  $\bigwedge^{(m)}(\mathcal{R}) \subseteq \mathrm{Cl}^{(m)}(V,\mathfrak{q})$ .
Remark E.31. For  $m \in [n]$  we use  $\rho^{(m)}$  to denote the group homomorphism  $\rho$  when restricted to act on the subvector space  $\mathrm{Cl}^{(m)}(V, \mathfrak{q})$ :
$$
\rho^ {(m)}: \Gamma (V, \mathfrak {q}) \rightarrow \mathrm {O} _ {\Lambda^ {(m)} (\mathcal {R})} \left(\operatorname {C l} ^ {(m)} (V, \mathfrak {q}), \bar {\mathfrak {q}}\right), \quad \rho^ {(m)} (w) := \rho (w) | _ {\operatorname {C l} ^ {(m)} (V, \mathfrak {q})}. \tag {525}
$$
By Theorem D.50 or Corollary E.22 we see that  $\bigwedge^{[ \times ]}(\mathcal{R})$  always lies inside the kernel of  $\rho^{(m)}$ :
$$
\bigwedge^ {[ \times ]} (\mathcal {R}) \subseteq \ker \rho^ {(m)} \subseteq \Gamma (V, \mathfrak {q}). \tag {526}
$$
So we get a well-defined group homorphism on the quotient:
$$
\bar {\rho} ^ {(m)}: \Gamma (V, \mathfrak {q}) / \bigwedge^ {[ \times ]} (\mathcal {R}) \rightarrow \mathrm {O} _ {\Lambda^ {(m)} (\mathcal {R})} \left(\operatorname {C l} ^ {(m)} (V, \mathfrak {q}), \bar {\mathfrak {q}}\right), \quad \bar {\rho} ^ {(m)} ([ w ]) := \rho (w) | _ {\operatorname {C l} ^ {(m)} (V, \mathfrak {q})}. \tag {527}
$$
Furthermore, by Theorem E.25 we have the isomorphism:
$$
\bar {\rho} ^ {(1)}: \Gamma (V, \mathfrak {q}) / \bigwedge^ {[ \times ]} (\mathcal {R}) \cong \mathrm {O} _ {\bigwedge^ {(1)} (\mathcal {R})} \left(\operatorname {C l} ^ {(1)} (V, \mathfrak {q}), \bar {\mathfrak {q}}\right) = \mathrm {O} _ {\mathcal {R}} (V, \mathfrak {q}), \quad \bar {\rho} ^ {(1)} ([ w ]) = \rho (w) | _ {V}. \tag {528}
$$
Consequently, for all  $m \in [n]$  we get the composition of group homorphisms:
$$
\tilde {\rho} ^ {(m)}: \mathrm {O} _ {\mathcal {R}} (V, \mathfrak {q}) \stackrel {(\bar {\rho} ^ {(1)}) ^ {- 1}} {\cong} \Gamma (V, \mathfrak {q}) / \bigwedge^ {[ \times ]} (\mathcal {R}) \stackrel {\bar {\rho} ^ {(m)}} {\rightarrow} \mathrm {O} _ {\Lambda^ {(m)} (\mathcal {R})} (\operatorname {C l} ^ {(m)} (V, \mathfrak {q}), \bar {\mathfrak {q}}), \tag {529}
$$
$$
\tilde {\rho} ^ {(m)} (\Phi) = \rho (w) | _ {\operatorname {C l} ^ {(m)} (V, \mathfrak {q})}, \quad f o r a n y w \in \Gamma (V, \mathfrak {q}) w i t h \rho^ {(1)} (w) = \Phi . \tag {530}
$$
Similarly, we get an (injective) group homomorphism:
$$
\tilde {\rho}: \mathrm {O} _ {\mathcal {R}} (V, \mathfrak {q}) \rightarrow \mathrm {O} _ {\Lambda (\mathcal {R})} (\operatorname {C l} (V, \mathfrak {q}), \bar {\mathfrak {q}}), \tag {531}
$$
$$
\tilde {\rho} (\Phi) := \rho (w), \quad f o r a n y w \in \Gamma (V, \mathfrak {q}) w i t h \rho^ {(1)} (w) = \Phi . \tag {532}
$$
So, the group  $\mathrm{O}_{\mathcal{R}}(V,\mathfrak{q})$  acts on  $\operatorname{Cl}(V,\mathfrak{q})$  and all subvector spaces  $\mathrm{Cl}^{(m)}(V,\mathfrak{q})$ ,  $m = 0,\dots,n$ , in the same way as  $\Gamma (V,\mathfrak{q})$  does via  $\rho$ , when using the surjective map  $\rho^{(1)}$  to lift elements  $\Phi \in \mathrm{O}_{\mathcal{R}}(V,\mathfrak{q})$  to elements  $w\in \Gamma (V,\mathfrak{q})$  with  $\rho^{(1)}(w) = \Phi$ .
Specifically, let  $x \in \mathrm{Cl}(V, \mathfrak{q})$  be of the form  $x = \sum_{i \in I} c_i \cdot v_{i,1} \cdots v_{i,k_i}$  with  $v_{i,j} \in V$ ,  $c_i \in \mathbb{F}$  and  $\Phi \in \mathrm{O}_{\mathcal{R}}(V, \mathfrak{q})$  is given by  $\Phi = \bar{\rho}^{(1)}([w])$  with  $w \in \Gamma(V, \mathfrak{q})$ , then we have:
$$
\rho (w) (x) = \sum_ {i \in I} c _ {i} \cdot \rho (w) \left(v _ {i, 1}\right) \dots \rho (w) \left(v _ {i, k _ {i}}\right) = \sum_ {i \in I} c _ {i} \cdot \Phi \left(v _ {i, 1}\right) \dots \Phi \left(v _ {i, k _ {i}}\right). \tag {533}
$$
This means that the action  $\tilde{\rho}$  of  $\mathrm{O}_{\mathcal{R}}(V,\mathfrak{q})$  transforms a multivector  $x$  by transforming all its vector components  $v_{i,j}$  by  $\Phi$ , acting through the usual orthogonal transformation on vectors.
As such, we have the following equivariance property with respect to  $\mathrm{O}_{\mathcal{R}}(V,\mathfrak{q})$ .
Corollary E.32. Let  $F(T_{1},\ldots ,T_{\ell +s})\in \mathbb{F}[T_{1},\ldots ,T_{\ell +s}]$  be a polynomial in  $\ell +s$  variables with coefficients in  $\mathbb{F}$  and let  $k\in \{0,\dots ,n\}$ . Further, consider  $\ell$  elements  $x_{1},\ldots ,x_{\ell}\in \mathrm{Cl}(V,\mathfrak{q})$  and  $s$  elements  $y_{1},\ldots ,y_{s}\in \bigwedge (\mathcal{R})$ . Then for every  $\Phi \in \mathrm{O}_{\mathcal{R}}(V,\mathfrak{q})$  we get the equivariance property:
$$
\tilde {\rho} (\Phi) \left(F \left(x _ {1},, \dots , x _ {\ell}, y _ {1}, \dots , y _ {s}\right) ^ {(k)}\right) = F \left(\tilde {\rho} (\Phi) \left(x _ {1}\right), \dots , \tilde {\rho} (\Phi) \left(x _ {\ell}\right), y _ {1}, \dots , y _ {s}\right) ^ {(k)}, \tag {534}
$$
E.5 The Spinor Norm and the Clifford Norm
In this subsection we shortly introduce the three slightly different versions of a norm that appear in the literature: the Spinor norm, the Clifford norm and the extended quadratic form. We are interested under which conditions do they have multiplicative behaviour.
Definition E.33 (The Spinor norm and the Clifford norm). We define the Spinor norm and the Clifford norm of  $\operatorname{Cl}(V, \mathfrak{q})$  as the maps:
$$
\operatorname {S N}: \operatorname {C l} (V, \mathfrak {q}) \rightarrow \operatorname {C l} (V, \mathfrak {q}), \quad \operatorname {S N} (x) := \beta (x) x, \tag {535}
$$
$$
\mathrm {C N}: \operatorname {C l} (V, \mathfrak {q}) \rightarrow \operatorname {C l} (V, \mathfrak {q}), \quad \mathrm {C N} (x) := \gamma (x) x. \tag {536}
$$
Also recall the extended quadratic form:
$$
\bar {\mathfrak {q}}: \operatorname {C l} (V, \mathfrak {q}) \rightarrow \mathbb {F}, \quad \bar {\mathfrak {q}} (x) = \zeta (\beta (x) x) = \zeta (\mathrm {S N} (x)). \tag {537}
$$
As a first preliminary Lemma we need to study when the projection onto the zero-component is multiplicative:
Lemma E.34. Let  $x \in \bigwedge(\mathcal{R})$  and  $y \in \operatorname{Cl}(V, \mathfrak{q})$  then we have:
$$
\zeta (x y) = \zeta (x) \zeta (y). \tag {538}
$$
As a result, the projection onto the zero component induces an  $\mathbb{F}$ -algebra homomorphism:
$$
\zeta : \bigwedge (\mathcal {R}) \rightarrow \mathbb {F}, \quad y \mapsto \zeta (y). \tag {539}
$$
Proof. We now use the notations from D.40 and Lemma D.41. We distinguish two cases:  $x \in \bigwedge^{(\geq 1)}(\mathcal{R})$  and:
$$
x \in \bigwedge (\mathcal {R}) \backslash \bigwedge^ {(\geq 1)} (\mathcal {R}) = \bigwedge^ {\times} (\mathcal {R}) = \mathbb {F} ^ {\times} + \bigwedge^ {(\geq 1)} (\mathcal {R}. \tag {540}
$$
In the first case, we have:  $\zeta(xy) = 0 = \zeta(x)\zeta(y)$ , as multiplying with  $x \in \bigwedge^{(\geq 1)}(\mathcal{R})$  can only increase the grade of occurring terms or make them vanish.
In the second case, we can write  $x = a + f$  with  $a \in \mathbb{F}^{\times}$  and  $f \in \bigwedge^{(\geq 1)}(\mathcal{R})$ . Clearly,  $\zeta(x) = a$ . We then get by linearity and the first case:
$$
\zeta (x y) = \zeta (a y + f y) = a \zeta (y) + \zeta (f y) = \zeta (x) \zeta (y) + 0. \tag {541}
$$
Lemma E.35. Let  $x_{1}, x_{2} \in \mathrm{Cl}(V, \mathfrak{q})$ .
1. If  $\mathrm{SN}(x_1) \in \mathfrak{Z}(\operatorname{Cl}(V, \mathfrak{q}))$  then we have:
$$
\mathrm {S N} \left(x _ {1} x _ {2}\right) = \mathrm {S N} \left(x _ {1}\right) \mathrm {S N} \left(x _ {2}\right). \tag {542}
$$
2. If  $\mathrm{CN}(x_1) \in \mathfrak{Z}(\operatorname{Cl}(V, \mathfrak{q}))$  then we have:
$$
\mathrm {C N} \left(x _ {1} x _ {2}\right) = \mathrm {C N} \left(x _ {1}\right) \mathrm {C N} \left(x _ {2}\right). \tag {543}
$$
3. If  $\mathrm{SN}(x_1) \in \bigwedge^{[0]}(\mathcal{R})$  or  $\mathfrak{q} = 0$  then we have:
$$
\bar {\mathfrak {q}} \left(x _ {1} x _ {2}\right) = \bar {\mathfrak {q}} \left(x _ {1}\right) \bar {\mathfrak {q}} \left(x _ {2}\right). \tag {544}
$$
Proof.  $\mathrm{SN}(x_1)\in \mathfrak{Z}(\mathrm{Cl}(V,\mathfrak{q}))$  implies:
$$
\operatorname {S N} \left(x _ {1} x _ {2}\right) = \beta \left(x _ {1} x _ {2}\right) x _ {1} x _ {2} \tag {545}
$$
$$
\begin{array}{l} = \beta \left(x _ {2}\right) \beta \left(x _ {1}\right) x _ {1} x _ {2} (546) \\ = \beta \left(x _ {2}\right) \operatorname {S N} \left(x _ {1}\right) x _ {2} (547) \\ \end{array}
$$
$$
\stackrel {\mathrm {S N} \left(x _ {1}\right) \in \underline {{3}} (\operatorname {C l} (V, \mathfrak {q}))} {=} \mathrm {S N} \left(x _ {1}\right) \beta \left(x _ {2}\right) x _ {2} \tag {548}
$$
$$
= \operatorname {S N} \left(x _ {1}\right) \operatorname {S N} \left(x _ {2}\right). \tag {549}
$$
Similarly for CN.
Together with Lemma E.34 and  $\mathrm{SN}(x_1)\in \bigwedge^{[0]}(\mathcal{R})\subseteq \bigwedge (\mathcal{R})\cap \mathfrak{Z}(\mathrm{Cl}(V,\mathfrak{q}))$  we get:
$$
\begin{array}{l} \bar {\mathbf {q}} \left(x _ {1} x _ {2}\right) = \zeta (\mathrm {S N} \left(x _ {1} x _ {2}\right)) (550) \\ = \zeta (\operatorname {S N} \left(x _ {1}\right) \operatorname {S N} \left(x _ {2}\right)) (551) \\ = \zeta (\mathrm {S N} (x _ {1})) \zeta (\mathrm {S N} (x _ {2})) (552) \\ = \bar {\mathfrak {q}} \left(x _ {1}\right) \bar {\mathfrak {q}} \left(x _ {2}\right). (553) \\ \end{array}
$$
Lemma E.36. Consider the following subset of  $\operatorname{Cl}(V, \mathfrak{q})$ :
$$
\Gamma^ {[ - ]} (V, \mathfrak {q}) := \left\{x \in \operatorname {C l} ^ {[ 0 ]} (V, \mathfrak {q}) \cup \operatorname {C l} ^ {[ 1 ]} (V, \mathfrak {q}) \mid \forall v \in V \exists v ^ {\prime} \in V. \alpha (x) v = v ^ {\prime} x \right\}. \tag {554}
$$
Then  $\Gamma^{[-]}(V,\mathfrak{q})$  is closed under multiplication and for every  $x\in \Gamma^{[-]}(V,\mathfrak{q})$  we have:
$$
\operatorname {S N} (x) \in \bigwedge^ {[ 0 ]} (\mathcal {R}), \quad \operatorname {C N} (x) \in \bigwedge^ {[ 0 ]} (\mathcal {R}). \tag {555}
$$
Proof. For  $x, y \in \Gamma^{[-]}(V, \mathfrak{q})$  we also have that  $xy$  is homogeneous. Furthermore, we get for  $v \in V$ :
$$
\begin{array}{l} \alpha (x y) v = \alpha (x) \alpha (y) v (556) \\ = \alpha (x) v ^ {\prime} y (557) \\ = \tilde {v} x y, (558) \\ \end{array}
$$
for some  $v'$ ,  $\tilde{v} \in V$ . So,  $xy \in \Gamma^{[-]}(V, \mathfrak{q})$  and  $\Gamma^{[-]}(V, \mathfrak{q})$  is closed under multiplication.
With the above conditions on  $x$  we get for every  $v \in V$ :
$$
\begin{array}{l} \alpha (\mathrm {S N} (x)) v = \alpha (\beta (x)) \alpha (x) v (559) \\ = \alpha (\beta (x)) v ^ {\prime} x (560) \\ = \alpha (\beta (x)) (- \alpha (\beta (v ^ {\prime}))) x (561) \\ = - \alpha (\beta (x) \beta \left(v ^ {\prime}\right)) x (562) \\ = - \alpha \left(\beta \left(v ^ {\prime} x\right)\right) x (563) \\ = - \alpha (\beta (\alpha (x) v)) x (564) \\ = - \beta (\alpha (\alpha (x) v)) x (565) \\ = - \beta (x \alpha (v)) x (566) \\ = \beta (x v) x (567) \\ = \beta (v) \beta (x) x (568) \\ = v \operatorname {S N} (x). (569) \\ \end{array}
$$
This implies by Theorem D.50 that:
$$
\operatorname {S N} (x) \in \bigwedge (\mathcal {R}). \tag {570}
$$
Since for homogeneous  $x \in \operatorname{Cl}(V, \mathfrak{q})$  we have:  $\operatorname{prt}(\beta(x)) = \operatorname{prt}(x)$  and thus:  $\operatorname{SN}(x) = \beta(x)x \in \operatorname{Cl}^{[0]}(V, \mathfrak{q})$ . This implies:
$$
\operatorname {S N} (x) \in \bigwedge (\mathcal {R}) \cap \operatorname {C l} ^ {[ 0 ]} (V, \mathfrak {q}) = \bigwedge^ {[ 0 ]} (\mathcal {R}). \tag {571}
$$
Theorem E.37 (Multiplicativity of the three different norms). Both, the Spinor norm and the Clifford norm, when restricted to the Clifford group, are well-defined group homomorphisms:
$$
\mathrm {S N}: \Gamma (V, \mathfrak {q}) \rightarrow \bigwedge^ {[ \times ]} (\mathcal {R}), \quad w \mapsto \mathrm {S N} (w) = \beta (w) w, \tag {572}
$$
$$
\mathrm {C N}: \Gamma (V, \mathfrak {q}) \rightarrow \bigwedge^ {[ \times ]} (\mathcal {R}), \quad w \mapsto \mathrm {C N} (w) = \gamma (w) w. \tag {573}
$$
Furthermore, the extended quadratic form  $\bar{\mathfrak{q}}$  of  $\operatorname{Cl}(V, \mathfrak{q})$  restricted to the Clifford group is a well-defined group homomorphism:
$$
\bar {\mathfrak {q}}: \Gamma (V, \mathfrak {q}) \rightarrow \mathbb {F} ^ {\times}, \quad w \mapsto \bar {\mathfrak {q}} (w) = \zeta (\beta (w) w). \tag {574}
$$
Proof. This directly follows from Lemma E.35 and Lemma E.36. Note that  $\Gamma(V, \mathfrak{q}) \subseteq \Gamma^{[-]}(V, \mathfrak{q})$ .
Example E.38. 1. For  $a\in \mathbb{F}$  we get:
$$
\operatorname {S N} (a) = \beta (a) a = a ^ {2}. \tag {575}
$$
This also shows:  $\bar{\mathfrak{q}}(a) = a^2\bar{\mathfrak{q}}(1) = a^2$ .
2. For  $w \in V$  we have:
$$
\operatorname {S N} (w) = \beta (w) w = w ^ {2} = \mathfrak {q} (w). \tag {576}
$$
This also shows:  $\bar{\mathfrak{q}}(w) = \mathfrak{q}(w)$ .
3. For  $\gamma = 1 + ef$  with  $e\in V$  and  $f\in \mathcal{R}$  we have: