SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \operatorname {S N} (\gamma) = (1 + f e) (1 + e f) (577) \\ = 1 + f e + e f + f e e f (578) \\ = 1 + 2 \mathfrak {b} (e, f) + \mathfrak {q} (f) \cdot \mathfrak {q} (e) (579) \\ = 1. (580) \\ \end{array}
$$
This also shows:  $\bar{\mathfrak{q}} (\gamma) = 1$
4. For  $g = 1 + h \in \bigwedge^{*}(\mathcal{R})$  we get:
$$
\begin{array}{l} \operatorname {S N} (g) = (1 + \beta (h)) (1 + h) (581) \\ = 1 + \beta (h) + h + \beta (h) h, (582) \\ \end{array}
$$
and thus:  $\bar{\mathfrak{q}}(g) = 1$ .
E.6 The Pin Group and the Spin Group
We have investigated the Clifford group and its action on the algebra through the twisted conjugation. In many fields of study, the Clifford group is further restricted to the Pin or Spin group. There can arise a few issues regarding the exact definition of these groups, especially when also considering fields other than the reals. We elaborate on these concerns here and leave the general definition for future discussion.
Motivation E.39 (The problem of generalizing the definition of the Spin group). For a positive definite quadratic form  $\mathfrak{q}$  on the real vector space  $V = \mathbb{R}^n$  with  $n \geq 3$  the Spin group  $\operatorname{Spin}(n)$  is defined via the kernel of the Spinor norm (=extended quadratic form on  $\mathrm{Cl}(V, \mathfrak{q})$ ) restricted to the special Clifford group  $\Gamma^{[0]}(V, \mathfrak{q})$ :
$$
\operatorname {S p i n} (n) := \ker \left(\bar {\mathfrak {q}}: \Gamma^ {[ 0 ]} (V, \mathfrak {q}) \rightarrow \mathbb {R} ^ {\times}\right) = \left\{w \in \Gamma^ {[ 0 ]} (V, \mathfrak {q}) \mid \bar {\mathfrak {q}} (w) = 1 \right\} = \bar {\mathfrak {q}} | _ {\Gamma^ {[ 0 ]} (V, \mathfrak {q})} ^ {- 1} (1). \tag {583}
$$
$\operatorname{Spin}(n)$  is thus a normal subgroup of the special Clifford group  $\Gamma^{[0]}(V, \mathfrak{q})$ , and, as it turns out, a double cover of the special orthogonal group  $\mathrm{SO}(n)$  via the twisted conjugation  $\rho$ . The latter can be summarized by the short exact sequence:
$$
1 \longrightarrow \left\{\pm 1 \right\} \xrightarrow {\text {i n c l}} \operatorname {S p i n} (n) \xrightarrow {\rho} \operatorname {S O} (n) \longrightarrow 1. \tag {584}
$$
We intend to generalize this in several directions: 1. from Spin to Pin group, 2. from  $\mathbb{R}^n$  to vector spaces  $V$  over general fields  $\mathbb{F}$  with  $\mathrm{char}(\mathbb{F})\neq 2$ , 3. from non-degenerate to degenerate quadratic forms  $\mathfrak{q}$ , 4. from positive (semi-)definite to non-definite quadratic forms  $\mathfrak{q}$ . This comes with several challenges and ambiguities.
If we want to generalize the above to define the Pin group we would allow for elements not just of (pure) even parity  $w \in \Gamma^{[0]}(V, \mathfrak{q})$ . Here the question arises if one should generalize to the unconstrained Clifford group  $\tilde{\Gamma}(V, \mathfrak{q})$  or the (homogeneous) Clifford group  $\Gamma(V, \mathfrak{q})$ . As discussed before, to ensure that the (adjusted) twisted conjugation  $\rho$  is a well-defined algebra automorphism of  $\operatorname{Cl}(V, \mathfrak{q})$  the parity homogeneity assumptions is crucial. Furthermore, the elements of  $\tilde{\Gamma}(V, \mathfrak{q})$  that lead to non-trivial orthogonal automorphisms, e.g.  $v \in V$  with  $\mathfrak{q}(v) \neq 0$  and  $\gamma = 1 + ef$  with  $e \in V$ ,  $f \in \mathcal{R}$ , are already homogeneous. So it is arguably safe and reasonable to restrict to the Clifford group  $\Gamma(V, \mathfrak{q})$  and define the Pin group  $\operatorname{Pin}(V, \mathfrak{q})$  as some subquotient of  $\Gamma(V, \mathfrak{q})$ .
In the non-definite (but still non-degenerate, real) case  $\mathbb{R}^{(p,q)}$ ,  $p, q \geq 1$ , the special orthogonal group  $\mathrm{SO}(p, q)$  contains combinations of reflections  $r_0 \circ r_1$  where the corresponding normal vectors  $v_0, v_1 \in V$  satisfy  $\mathfrak{q}(v_0) = 1$  and  $\mathfrak{q}(v_1) = -1$ . Their product  $v_0 v_1$  would lie in the special Clifford group  $\Gamma^{[0]}(V, \mathfrak{q})$ . However, their spinor norm would be different from 1:
$$
\bar {\mathfrak {q}} \left(v _ {0} v _ {1}\right) = \mathfrak {q} \left(v _ {0}\right) \mathfrak {q} \left(v _ {1}\right) = - 1 \neq 1. \tag {585}
$$
Here now the question arises if we would like to preserve the former definition of  $\mathrm{Spin}(p,q)$  as  $\overline{\mathfrak{q}}|_{\Gamma^{[0]}(V,\mathfrak{q})}^{-1}(1)$  and exclude  $v_0 v_1$  from  $\mathrm{Spin}(p,q)$ , or, if we adjust the definition of  $\mathrm{Spin}(p,q)$  and include  $v_0 v_1$ . The former definition has the effect that  $\mathrm{Spin}(p,q)$  does, in general, not map surjectively onto  $\mathrm{SO}(p,q)$  anymore, and we would only get a short exact sequence:
$$
1 \longrightarrow \{\pm 1 \} \xrightarrow {\text {i n c l}} \operatorname {S p i n} (p, q) \xrightarrow {\rho} \operatorname {S O} (p, q) \xrightarrow {\bar {\bar {q}}} \underbrace {\mathbb {R} ^ {\times} / (\mathbb {R} ^ {\times}) ^ {2}} _ {\cong \{\pm 1 \}}. \tag {586}
$$
The alternative would be to define  $\mathrm{Spin}(p,q)$  as  $\bar{\mathfrak{q}}|_{\Gamma^{[0]}(V,\mathfrak{q})}^{-1}(\pm 1)$ . This would allow for  $v_0 v_1 \in \mathrm{Spin}(p,q)$  and lead to the short exact sequence:
$$
1 \longrightarrow \underbrace {\mu_ {4} (\mathbb {R})} _ {= \{\pm 1 \}} \stackrel {\text {i n c l}} {\longrightarrow} \operatorname {S p i n} (p, q) \stackrel {\rho} {\longrightarrow} \operatorname {S O} (p, q) \longrightarrow 1, \tag {587}
$$
which exactly recovers the former behaviour for  $\operatorname{Spin}(n)$ , and, which makes  $\operatorname{Spin}(p,q)$  a double cover of  $\mathrm{SO}(p,q)$ .
However, for other fields  $\mathbb{F}$ ,  $\mathrm{char}(\mathbb{F}) \neq 2$ , and non-degenerate  $(V,\mathfrak{q})$ ,  $\dim V \geq 3$ , one would get, with the last definition  $\bar{\mathfrak{q}}|_{\Gamma^{[0]}(V,\mathfrak{q})}^{-1}(\pm 1)$ , the exact sequence:
$$
1 \longrightarrow \mu_ {4} (\mathbb {F}) \xrightarrow {\text {i n c l}} \operatorname {S p i n} (V, \mathfrak {q}) \xrightarrow {\rho} \operatorname {S O} (V, \mathfrak {q}) \xrightarrow {\bar {\bar {\mathfrak {q}}}} \mathbb {F} ^ {\times} / \left(\mathbb {F} ^ {\times}\right) ^ {2}, \tag {588}
$$
and one would need to live with the fact that: a)  $\mu_4(\mathbb{F})\coloneqq \left\{x\in \mathbb{F}^\times \mid x^4 = 1\right\}$  could contain a number of elements  $k$  different from 2, rendering  $\rho$  a  $(k:1)$ -map, in contrast to a  $(2:1)$ -map, and, again, b) the non-surjectivity of  $\rho$ . The latter comes from the fact that a combination of reflections  $r_0\circ r_1$  where the corresponding normal vectors  $v_{0},v_{1}\in V$  with  $\mathfrak{q}(v_0),\mathfrak{q}(v_1)\neq 0$  cannot, in general, be normalized such that the  $\mathfrak{q}(v_i)$ 's lie in  $\{\pm 1\}$  by multiplying/dividing the  $v_{i}$ 's with some scalars  $c_{i}\in \mathbb{F}^{\times}$ . Note that we get:
$$
\mathfrak {q} \left(v _ {i} / c _ {i}\right) = \mathfrak {q} \left(v _ {i}\right) / c _ {i} ^ {2}. \tag {589}
$$
This shows that we can only normalize the  $v_{i}$ 's such that the  $\mathfrak{q}(v_i)$ 's lie inside a fixed system of representatives  $S \subseteq \mathbb{F}^{\times}$  of  $\mathbb{F}^{\times} / (\mathbb{F}^{\times})^{2}$ , which is thus of size:
$$
\# S = \# \left(\mathbb {F} ^ {\times} / \left(\mathbb {F} ^ {\times}\right) ^ {2}\right), \tag {590}
$$
which can be different from  $2 = \# \{\pm 1\}$
So, in this general setting, the first definition of  $\mathrm{Spin}(V,\mathfrak{q})$  as  $\bar{\mathfrak{q}}|_{\Gamma^{[0]}(V,\mathfrak{q})}^{-1}(1)$  would at least correct the map  $\rho$  to be a  $(2:1)$ -map. We would get the following short exact seequence:
$$
1 \longrightarrow \{\pm 1 \} \stackrel {\text {i n c l}} {\longrightarrow} \operatorname {S p i n} (V, \mathfrak {q}) \stackrel {\rho} {\longrightarrow} \operatorname {S O} (V, \mathfrak {q}) \stackrel {\bar {\bar {\mathfrak {q}}}} {\longrightarrow} \mathbb {F} ^ {\times} / (\mathbb {F} ^ {\times}) ^ {2}. \tag {591}
$$
The normalization argument around Equation 589 for general fields  $\mathbb{F}$  now would also give us a third option: we could, instead of restricting elements to have a fixed value  $\mathfrak{q}(v_i)\in S$ , which depends on the choice of  $S$ , identify elements  $w_{1},w_{2}\in \Gamma^{[0]}(V,\mathfrak{q})$  if they differ by a scalar  $c\in \mathbb{F}^{\times}$ :
$$
w _ {1} = c \cdot w _ {2}. \tag {592}
$$
Then for their spinor norms (modulo  $(\mathbb{F}^{\times})^{2}$ ) we would get:
$$
\bar {\mathfrak {q}} \left(w _ {1}\right) = c ^ {2} \cdot \bar {\mathfrak {q}} \left(w _ {2}\right), \quad \left[ \bar {\mathfrak {q}} \left(w _ {1}\right) \right] = \left[ \bar {\mathfrak {q}} \left(w _ {2}\right) \right] \in \mathbb {F} ^ {\times} / \left(\mathbb {F} ^ {\times}\right) ^ {2}. \tag {593}
$$
So, the spinor norms of  $w_{1}$  and  $w_{2}$  would be represented by the same representative  $s \in S$ , as desired. However, this definition would just identify  $\operatorname{Spin}(V, \mathfrak{q})$  with  $\mathrm{SO}(V, \mathfrak{q})$  via  $\rho$ :
$$
\operatorname {S p i n} (V, \mathfrak {q}) = \Gamma^ {[ 0 ]} (V, \mathfrak {q}) / \mathbb {F} ^ {\times} \stackrel {\rho} {\cong} \operatorname {S O} (V, \mathfrak {q}), \tag {594}
$$
and nothing new would emerge from this. Note that the latter isomorphisms always holds for non-degenerate  $(V, \mathfrak{q})$  with  $\dim V \geq 3$ ,  $\operatorname{char}(\mathbb{F}) \neq 2$ , and can be expressed as the exact sequence:
$$
1 \longrightarrow \mathbb {F} ^ {\times} \xrightarrow {\operatorname {i n c l}} \Gamma^ {[ 0 ]} (V, \mathfrak {q}) \xrightarrow {\rho} \mathrm {S O} (V, \mathfrak {q}) \longrightarrow 1. \tag {595}
$$
A fourth option would be to mod out the scalar squares  $(\mathbb{F}^{\times})^{2}$  instead of  $\mathbb{F}^{\times}$  and use  $\Gamma^{[0]}(V,\mathfrak{q}) / (\mathbb{F}^{\times})^{2}$  as the definition of  $\mathrm{Spin}(V,\mathfrak{q})$ . This would lead to the exact sequence:
$$
1 \longrightarrow \mathbb {F} ^ {\times} / (\mathbb {F} ^ {\times}) ^ {2} \xrightarrow {\text {i n c l}} \operatorname {S p i n} (V, \mathfrak {q}) \xrightarrow {\rho} \operatorname {S O} (V, \mathfrak {q}) \longrightarrow 1, \tag {596}
$$
which would again coincide with the real case of  $\operatorname{Spin}(n)$  as then  $\mathbb{R}^{\times} / (\mathbb{R}^{\times})^{2} \cong \{\pm 1\}$ . However, in the general case, again,  $\rho$  is here a  $(k:1)$ -map instead of a  $(2:1)$ -map with  $k := \# \left( \mathbb{F}^{\times} / (\mathbb{F}^{\times})^{2} \right)$ . Furthermore, the spinor norm  $\bar{\mathfrak{q}}$  on  $\operatorname{Spin}(V,\mathfrak{q})$  would not be well-defined anymore, in its current form, as different representatives of elements  $[w_1] = [w_2] \in \Gamma^{[0]}(V,\mathfrak{q}) / (\mathbb{F}^{\times})^{2}$  would differ by a scalar square:  $w_{1} = c^{2} \cdot w_{2}$ . Their spinor norms would thus differ by by a forth scalar power:
$$
\bar {\mathfrak {q}} \left(w _ {1}\right) = \bar {\mathfrak {q}} \left(c ^ {2} \cdot w _ {2}\right) = c ^ {4} \cdot \bar {\mathfrak {q}} \left(w _ {2}\right). \tag {597}
$$
So, the spinor norm on  $\Gamma^{[0]}(V,\mathfrak{q}) / (\mathbb{F}^{\times})^{2}$  would only be well defined modulo  $(\mathbb{F}^{\times})^{4}\subseteq (\mathbb{F}^{\times})^{2}$ :
$$
[ \bar {\mathfrak {q}} ]: \Gamma^ {[ 0 ]} (V, \mathfrak {q}) / (\mathbb {F} ^ {\times}) ^ {2} \rightarrow \mathbb {F} ^ {\times} / (\mathbb {F} ^ {\times}) ^ {4}, \quad [ w ] \mapsto [ \bar {\mathfrak {q}} (w) ]. \tag {598}
$$
Things become even more complicated in the degenerate case. At least we always have a short exact sequence for  $m \geq 3$ :
$$
1 \longrightarrow \bigwedge^ {[ \times ]} (\mathcal {R}) \xrightarrow {\text {i n c l}} \Gamma^ {[ 0 ]} (V, \mathfrak {q}) \xrightarrow {\rho} \mathrm {S O} _ {\mathcal {R}} (V, \mathfrak {q}) \longrightarrow 1, \tag {599}
$$
where  $\mathrm{SO}_{\mathcal{R}}(V,\mathfrak{q})$  indicates the set of those special orthogonal automorphisms  $\Phi$  of  $(V,\mathfrak{q})$  with  $\Phi|_{\mathcal{R}} = \mathrm{id}_{\mathcal{R}}$ , where  $\mathcal{R}$  is the radical subspace of  $(V,\mathfrak{q})$  with  $r \coloneqq \dim \mathcal{R}$ ,  $n \coloneqq \dim V$ ,  $m \coloneqq n - r$ . Recall that we have:
$$
\bigwedge^ {[ \times ]} (\mathcal {R}) = \mathbb {F} ^ {\times} \cdot \bigwedge^ {[ * ]} (\mathcal {R}), \tag {600}
$$
$$
\bigwedge^ {[ * ]} (\mathcal {R}) = 1 + \operatorname {s p a n} \left\{f _ {1} \dots f _ {k} \mid k \geq 2 \text {e v e n}, f _ {l} \in \mathcal {R}, l = 1, \dots , k \right\}. \tag {601}
$$
Note that for  $g \in \bigwedge^{[*]}(\mathcal{R})$  we have  $\rho(g)|_V = \mathrm{id}_V$  and  $\bar{\mathfrak{q}}(g) = 1$ . So the elements from  $\bigwedge^{[*]}(\mathcal{R})$  are only blowing up the kernels of  $\rho$  and  $\bar{\mathfrak{q}}$  and can be considered redundant for our analysis. So one can argue that one can mod out  $\bigwedge^{[*]}(\mathcal{R})$  in the above groups. We thus get a short exact sequence:
$$
1 \longrightarrow \mathbb {F} ^ {\times} \longrightarrow \underbrace {\Gamma^ {[ 0 ]} (V , \mathfrak {q}) / \bigwedge^ {[ * ]} (\mathcal {R})} _ {=: \tilde {\Gamma} ^ {[ 0 ]} (V, \mathfrak {q})} \xrightarrow {\rho} \mathrm {S O} _ {\mathcal {R}} (V, \mathfrak {q}) \longrightarrow 1, \tag {602}
$$
which now looks similar to the non-degenerate case. We can now consider the same 4 options for the definition of the Spin group as before:
$$
\bar {\mathbf {q}} | _ {\tilde {\Gamma} ^ {[ 0 ]} (V, \mathfrak {q})} ^ {- 1} (1), \quad \bar {\mathbf {q}} | _ {\tilde {\Gamma} ^ {[ 0 ]} (V, \mathfrak {q})} ^ {- 1} (\pm 1), \quad \tilde {\Gamma} ^ {[ 0 ]} (V, \mathfrak {q}) / \mathbb {F} ^ {\times}, \quad \tilde {\Gamma} ^ {[ 0 ]} (V, \mathfrak {q}) / (\mathbb {F} ^ {\times}) ^ {2}. \tag {603}
$$
As before, the third option can easily be discarded. If we want to preserve generality, we have the option to either preserve  $\bar{\mathfrak{q}}$  and the  $(2:1)$ -property of  $\rho$  and pick the first option, or, preserve the surjectivity of  $\rho$  and take the fourth option. If we are only interested in the  $\mathbb{R}$ -case, then the second option preserves all properties. Note that in the  $\mathbb{R}$ -case the groups of the second and forth option are isomorphic as groups anyways:
$$
\bar {\mathfrak {q}} | _ {\tilde {\Gamma} ^ {[ 0 ]} (V, \mathfrak {q})} ^ {- 1} (\pm 1) \cong \tilde {\Gamma} ^ {[ 0 ]} (V, \mathfrak {q}) / \left(\mathbb {R} ^ {\times}\right) ^ {2}. \tag {604}
$$
The reason is that we always have that:  $\bar{\mathfrak{q}} (\mathbb{F}^{\times}) = (\mathbb{F}^{\times})^{2}$ , and, in the  $\mathbb{R}$ -case, the left group already contains the relevant elements to also map surjectively onto  $\mathrm{SO}_{\mathcal{R}}(V,\mathfrak{q})$  via  $\rho$ .
One could further discuss if one wanted to replace the extended quadratic form  $\bar{\mathfrak{q}}$ , which is given by  $\bar{\mathfrak{q}}(x) = \zeta(\beta(x)x) \in \mathbb{F}^{\times}$ , by the other possible definition of the spinor norm SN, which is only given by  $\mathrm{SN}(x) = \beta(x)x \in \bigwedge^{[\times]}(\mathcal{R})$ . However, for all relevant elements of  $\Gamma^{[0]}(V,\mathfrak{q})$  both definitions agree, but the description of the set of the rather irrelevant elements, which satisfy  $\mathrm{SN}(x) = 1$  and  $\rho(x)|_V = \mathrm{id}_V$ , becomes more complicated than the set  $\bigwedge^{[\times]}(\mathcal{R})$ . As we mod those irrelevant elements out anyways and we prefer to have our "norm map" to map to the scalars  $\mathbb{F}^{\times}$  instead of to  $\bigwedge^{[\times]}(\mathcal{R})$ , it is safe and reasonable to work with the extended quadratic form  $\bar{\mathfrak{q}}$  in all definitions.
In this paper we are mostly interested in working with the orthogonal groups and thus are interested in preserving the surjectivity of  $\rho$ . Since, for computational reasons, we usually restrict ourselves to the  $\mathbb{R}$ -case, it is easier to work with (a restricted set of) elements of a group than equivalence classes, and, the spinor norm/extended quadratic form has computational meaning, we side with the second definition in this paper, but only state it for the  $\mathbb{R}$ -case below. We leave the general definition for future discussion.
Definition E.40 (The real Pin group and the real Spin group). Let  $V$  be a finite dimensional  $\mathbb{R}$ -vector space  $V$ ,  $\dim V = n < \infty$ , and  $\mathfrak{q}$  a (possibly degenerate) quadratic form on  $V$ . We define the (real) Pin group and (real) Spin group, resp., of  $(V, \mathfrak{q})$  as the following subquotients of the Clifford group  $\Gamma(V, \mathfrak{q})$  and its even parity part  $\Gamma^{[0]}(V, \mathfrak{q})$ , resp.:
$$
\operatorname {P i n} (V, \mathfrak {q}) := \{x \in \Gamma (V, \mathfrak {q}) \mid \bar {\mathfrak {q}} (x) \in \{\pm 1 \} \} / \bigwedge^ {[ * ]} (\mathcal {R}), \tag {605}
$$
$$
\operatorname {S p i n} (V, \mathfrak {q}) := \left\{x \in \Gamma^ {[ 0 ]} (V, \mathfrak {q}) \mid \bar {\mathfrak {q}} (x) \in \{\pm 1 \} \right\} / \bigwedge^ {[ * ]} (\mathcal {R}). \tag {606}
$$
If  $(V,\mathfrak{q}) = \mathbb{R}^{(p,q,r)}$  is the standard quadratic  $\mathbb{R}$ -vector space with signature  $(p,q,r)$  then we denote:
$$
\operatorname {P i n} (p, q, r) := \operatorname {P i n} \left(\mathbb {R} ^ {(p, q, r)}\right), \tag {607}
$$
$$
\operatorname {S p i n} (p, q, r) := \operatorname {S p i n} \left(\mathbb {R} ^ {(p, q, r)}\right). \tag {608}
$$
Corollary E.41. Let  $(V, \mathfrak{q})$  be a finite dimensional quadratic vector space over  $\mathbb{R}$ . Then the twisted conjugation induces a well-defined and surjective group homomorphism onto the group of radical preserving orthogonal automorphisms of  $(V, \mathfrak{q})$ :
$$
\rho : \operatorname {P i n} (V, \mathfrak {q}) \rightarrow \mathrm {O} _ {\mathcal {R}} (V, \mathfrak {q}), \tag {609}
$$
with kernel:
$$
\ker (\rho : \operatorname {P i n} (V, \mathfrak {q}) \rightarrow \mathrm {O} _ {\mathcal {R}} (V, \mathfrak {q})) = \{\pm 1 \}. \tag {610}
$$
Correspondingly, for the  $\mathrm{Spin}(V,\mathfrak{q})$  group. In short, we have short exact sequences:
$$
1 \longrightarrow \left\{\pm 1 \right\} \stackrel {\text {i n c l}} {\longrightarrow} \operatorname {P i n} (V, \mathfrak {q}) \stackrel {\rho} {\longrightarrow} \mathrm {O} _ {\mathcal {R}} (V, \mathfrak {q}) \longrightarrow 1, \tag {611}
$$
$$
1 \longrightarrow \{\pm 1 \} \xrightarrow {\text {i n c l}} \operatorname {S p i n} (V, \mathfrak {q}) \xrightarrow {\rho} \operatorname {S O} _ {\mathcal {R}} (V, \mathfrak {q}) \longrightarrow 1. \tag {612}
$$
The examples E.19, E.20, E.21, E.38 allow us to describe the elements of the Pin and Spin group as follows.
Corollary E.42 (The elements of the real Pin group and the real Spin group). Let  $(V, \mathfrak{q})$  be a finite dimensional quadratic vector space over  $\mathbb{R}$  with signature  $(p, q, r)$ . We get the following description of the elements of the Pin group:
$$
\operatorname {P i n} (V, \mathfrak {q}) = \left\{\pm v _ {1} \dots v _ {k} \cdot \gamma_ {1} \dots \gamma_ {p + q} \cdot g \mid k \in \mathbb {N} _ {0}, \mathfrak {q} (v _ {l}) \in \{\pm 1 \}, g \in \bigwedge^ {[ * ]} (\mathcal {R}) \right\} / \bigwedge^ {[ * ]} (\mathcal {R}), \tag {613}
$$
where  $\gamma_{i} = 1 + e_{i}\sum_{j = 1}^{r}c_{i,j}e_{p + q + j}, c_{i,j}\in \mathbb{R},$  for  $i = 1,\dots ,p + q$  and  $j = 1,\ldots ,r$ , and,  $v_{l}\in V$  with  $\mathfrak{q}(v_l)\in \{\pm 1\}$  with  $l = 1,\dots ,l$  and  $k\in \mathbb{N}_0$ . Note that  $\{e_{p + q + j}\mid j\in [r]\}$  is meant to span the radical subspace  $\mathcal{R}$  of  $(V,\mathfrak{q})$ .
We similarly can describe the Spin group as follows:
$$
\operatorname {S p i n} (V, \mathfrak {q}) = \left\{\pm v _ {1} \dots v _ {k} \cdot \gamma_ {1} \dots \gamma_ {p + q} \cdot g \mid k \in 2 \mathbb {N} _ {0}, \mathfrak {q} (v _ {l}) \in \{\pm 1 \}, g \in \bigwedge^ {[ * ]} (\mathcal {R}) \right\} / \bigwedge^ {[ * ]} (\mathcal {R}), \tag {614}
$$
with the same conditions as above, but where  $k$  needs to be an even number.
Corollary E.43. Note that we also get well-defined group representations:
$$
\rho : \operatorname {P i n} (V, \mathfrak {q}) \rightarrow \operatorname {A u t} _ {\mathbf {A l g}, \operatorname {g r d}} (\operatorname {C l} (V, \mathfrak {q})) \cap \mathrm {O} _ {\Lambda (\mathcal {R})} (\operatorname {C l} (V, \mathfrak {q}), \bar {\mathfrak {q}}), \tag {615}
$$
with kernel  $\ker \rho = \{\pm 1\}$
In particular,  $\operatorname{Cl}(V, \mathfrak{q})$  and  $\operatorname{Cl}^{(m)}(V, \mathfrak{q})$  for  $m = 0, \dots, n$ , are orthogonal group representations of  $\operatorname{Pin}(V, \mathfrak{q})$  via  $\rho$ :
$$
\rho : \operatorname {P i n} (V, \mathfrak {q}) \rightarrow \mathrm {O} _ {\Lambda^ {(m)} (\mathcal {R})} \left(\mathrm {C l} ^ {(m)} (V, \mathfrak {q}), \bar {\mathfrak {q}}\right). \tag {616}
$$
Also, if  $F(T_{1},\ldots ,T_{\ell +s})\in \mathbb{R}[T_{1},\ldots ,T_{\ell +s}]$  is a polynomial in  $\ell +s$  variables with coefficients in  $\mathbb{R}$  and  $x_{1},\ldots ,x_{\ell}\in \mathrm{Cl}(V,\mathfrak{q})$ ,  $y_{1},\ldots ,y_{s}\in \bigwedge (\mathcal{R})$ , and  $k\in \{0,\dots ,n\}$ . Then for every  $w\in \operatorname {Pin}(V,\mathfrak{q})$  we have the equivariance property:
$$
\rho (w) \left(F \left(x _ {1}, \dots , x _ {\ell}, y _ {1}, \dots , y _ {s}\right) ^ {(k)}\right) = F (\rho (w) (x _ {1}), \dots , \rho (w) (x _ {\ell}), y _ {1}, \dots , y _ {s}) ^ {(k)}. \tag {617}
$$