SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \left(\rho \left(w _ {2}\right) \circ \rho \left(w _ {1}\right)\right) (x) = \rho \left(w _ {2}\right) \left(\rho \left(w _ {1}\right) (x)\right) (376) \\ = \rho \left(w _ {2}\right) \left(w _ {1} \alpha^ {\operatorname {p r t} \left(w _ {1}\right)} (x) w _ {1} ^ {- 1}\right) (377) \\ = w _ {2} \alpha^ {\operatorname {p r t} \left(w _ {2}\right)} \left(w _ {1} \alpha^ {\operatorname {p r t} \left(w _ {1}\right)} (x) w _ {1} ^ {- 1}\right) w _ {2} ^ {- 1} (378) \\ = w _ {2} \alpha^ {\operatorname {p r t} \left(w _ {2}\right)} \left(w _ {1}\right) \alpha^ {\operatorname {p r t} \left(w _ {2}\right)} \left(\alpha^ {\operatorname {p r t} \left(w _ {1}\right)} (x)\right) \alpha^ {\operatorname {p r t} \left(w _ {2}\right)} \left(w _ {1}\right) ^ {- 1} w _ {2} ^ {- 1} (379) \\ = w _ {2} (- 1) ^ {\operatorname {p r t} \left(w _ {2}\right) \operatorname {p r t} \left(w _ {1}\right)} w _ {1} \alpha^ {\operatorname {p r t} \left(w _ {2}\right) + \operatorname {p r t} \left(w _ {1}\right)} (x) (- 1) ^ {\operatorname {p r t} \left(w _ {2}\right) \operatorname {p r t} \left(w _ {1}\right)} w _ {1} ^ {- 1} w _ {2} ^ {- 1} (380) \\ = w _ {2} w _ {1} \alpha^ {\operatorname {p r t} \left(w _ {2}\right) + \operatorname {p r t} \left(w _ {1}\right)} (x) w _ {1} ^ {- 1} w _ {2} ^ {- 1} (381) \\ = \left(w _ {2} w _ {1}\right) \alpha^ {\operatorname {p r t} \left(w _ {2} w _ {1}\right)} (x) \left(w _ {2} w _ {1}\right) ^ {- 1} (382) \\ = \rho \left(w _ {2} w _ {1}\right) (x), (383) \\ \end{array}
$$
where we used the multiplicativity of  $\alpha^{\mathrm{prt}(w)}(x)$ :
$$
\begin{array}{l} \alpha^ {w} (x) \alpha^ {w} (y) = \left((- 1) ^ {\operatorname {p r t} (w)} x ^ {[ 1 ]} + x [ 0 ]\right) \left((- 1) ^ {\operatorname {p r t} (w)} y ^ {[ 1 ]} + y ^ {[ 0 ]}\right) (384) \\ = (- 1) ^ {\operatorname {p r t} (w)} \left(x ^ {[ 0 ]} y ^ {[ 1 ]} + x ^ {[ 1 ]} y ^ {[ 0 ]}\right) + x ^ {[ 1 ]} y ^ {[ 1 ]} + x ^ {[ 0 ]} y ^ {[ 0 ]} (385) \\ = \alpha^ {w} ((x y) ^ {[ 1 ]}) + (x y) ^ {[ 0 ]} (386) \\ = \alpha^ {w} (x y). (387) \\ \end{array}
$$
This implies:
$$
\rho \left(w _ {2}\right) \circ \rho \left(w _ {1}\right) = \rho \left(w _ {2} w _ {1}\right), \tag {388}
$$
which shows the claim.
Finally, we want to re-check that our newly defined  $\rho$ , despite its different appearance, still has the proper interpretation of a reflection.
Remark E.11. Let  $w, v \in V$  with  $\mathfrak{q}(w) \neq 0$ . Then  $\rho(w)(v)$  is the reflection of  $v$  w.r.t. the hyperplane that is normal to  $w$ :
$$
\rho (w) (v) = w \alpha^ {w} (v) w ^ {- 1} = - w v w ^ {- 1} = r _ {w} (v). \tag {389}
$$
Remark E.12. The presented results in this subsection can be slightly generalized as follows. If  $w \in \mathrm{Cl}^{\times}(V, \mathfrak{q})$  such that  $\eta(w) \in \bigwedge(\mathcal{R})$  then by Theorem D.50 we get for all  $v \in V$ :
$$
v \eta (w) = \alpha (\eta (w)) v = \eta (w) ^ {- 1} v. \tag {390}
$$
This implies that for all  $v \in V$ :
$$
\eta (w) v \eta (w) v = \eta (w) \eta (w) ^ {- 1} v v = \mathfrak {q} (v), \tag {391}
$$
and thus for all  $v \in V$ :
$$
(\alpha (w) v w ^ {- 1}) (\alpha (w) v w ^ {- 1}) = \mathfrak {q} (v). \tag {392}
$$
By the universal property of the Clifford algebra the map:
$$
\rho (w): V \rightarrow \operatorname {C l} (V, \mathfrak {q}), \quad v \mapsto \alpha (w) v w ^ {- 1}, \tag {393}
$$
uniquely extends to an  $\mathbb{F}$ -algebra homomorphism:
$$
\rho (w): \operatorname {C l} (V, \mathfrak {q}) \rightarrow \operatorname {C l} (V, \mathfrak {q}), \tag {394}
$$
$$
\begin{array}{l} x = c _ {0} + \sum_ {i \in I} c _ {i} \cdot v _ {i, 1} \dots v _ {i, k _ {i}} (395) \\ \mapsto c _ {0} + \sum_ {i \in I} c _ {i} \cdot \alpha (w) v _ {i, 1} w ^ {- 1} \dots \alpha (w) v _ {i, k _ {i}} w ^ {- 1} (396) \\ = w \left(c _ {0} + \sum_ {i \in I} c _ {i} \cdot \eta (w) v _ {i, 1} \dots \eta (w) v _ {i, k _ {i}}\right) w ^ {- 1} (397) \\ = w \left(c _ {0} + \sum_ {\substack {i \in I \\ k _ {i} \text {even}}} c _ {i} \cdot v _ {i, 1} \dots v _ {i, k _ {i}} + \eta (w) \cdot \sum_ {\substack {i \in I \\ k _ {i} \text {odd}}} c _ {i} \cdot v _ {i, 1} \dots v _ {i, k _ {i}}\right) w ^ {- 1} (398) \\ = w \left(x ^ {[ 0 ]} + \eta (w) \cdot x ^ {[ 1 ]}\right) w ^ {- 1}. (399) \\ \end{array}
$$
To further ensure that the elements  $w \in \mathrm{Cl}^{\times}(V, \mathfrak{q})$  with the above property form a group we might need to further restrict to require that  $\eta(w) \in \bigwedge^{[\times]}(\mathcal{R})$ . At least in this case we get for  $w_1, w_2 \in \mathrm{Cl}^{\times}(V, \mathfrak{q})$  with  $\eta(w_1), \eta(w_2) \in \bigwedge^{[\times]}(\mathcal{R}) \subseteq \mathfrak{Z}(\mathrm{Cl}(V, \mathfrak{q}))$ , see Remark D.43, that:
$$
\eta \left(w _ {2} w _ {1}\right) = w _ {1} ^ {- 1} \eta \left(w _ {2}\right) w _ {1} \eta \left(w _ {1}\right) = w _ {1} ^ {- 1} w _ {1} \eta \left(w _ {2}\right) \eta \left(w _ {1}\right) = \eta \left(w _ {2}\right) \eta \left(w _ {1}\right) \in \bigwedge^ {[ \times ]} (\mathcal {R}). \tag {400}
$$
So the following set:
$$
C := \left\{w \in \mathrm {C l} ^ {\times} (V, \mathfrak {q}) \mid \eta (w) \in \bigwedge^ {[ \times ]} (\mathcal {R}) \right\} \tag {401}
$$
is a subgroup of  $\mathrm{Cl}^{\times}(V,\mathfrak{q})$  where every  $w\in C$  defines an algebra homomorphism:
$$
\rho (w): \operatorname {C l} (V, \mathfrak {q}) \rightarrow \operatorname {C l} (V, \mathfrak {q}), \quad \rho (w) (x) = w \left(x ^ {[ 0 ]} + \eta (w) x ^ {[ 1 ]}\right) w ^ {- 1}. \tag {402}
$$
E.2 The Clifford Group
Motivation E.13. We have seen in the last section that if we choose homogeneous invertible elements  $w \in \mathrm{Cl}^{[\times]}(V, \mathfrak{q})$  then the action  $\rho(w)$ , the (adjusted) twisted conjugation, is an algebra automorphism of  $\operatorname{Cl}(V, \mathfrak{q})$  that also preserves the parity grading of  $\operatorname{Cl}(V, \mathfrak{q})$ , in particular,  $\rho(w)$  is linear and multiplicative.
We now want to investigate under which conditions on  $w$  the algebra automorphism  $\rho(w)$  also preserves the multivector grading:
$$
\operatorname {C l} (V, \mathfrak {q}) = \operatorname {C l} ^ {(0)} (V, \mathfrak {q}) \oplus \operatorname {C l} ^ {(1)} (V, \mathfrak {q}) \oplus \dots \oplus \operatorname {C l} ^ {(m)} (V, \mathfrak {q}) \oplus \dots \oplus \operatorname {C l} ^ {(n)} (V, \mathfrak {q}). \tag {403}
$$
If this was the case then each component  $\mathrm{Cl}^{(m)}(V,\mathfrak{q})$  would give rise to a corresponding group representation.
To preserve the multivector grading we at least need that for  $v \in V = \mathrm{Cl}^{(1)}(V, \mathfrak{q})$  we have that also  $\rho(w)(v) \in \mathrm{Cl}^{(1)}(V, \mathfrak{q}) = V$ . We will see that for  $w \in \mathrm{Cl}^{\times}(V, \mathfrak{q})$  such that  $\rho(w)$  is an algebra homomorphism, i.e. for  $w$  homogeneous and invertible, and such that  $\rho(w)(v) \in V$  for all  $v \in V$ , we already get the preservation of the whole multivector grading of  $\mathrm{Cl}(V, \mathfrak{q})$ .
Again, let  $(V,\mathfrak{q})$  be a quadratic space over a field  $\mathbb{F}$  with  $\mathrm{char}(\mathbb{F})\neq 2$  and  $\dim V = n < \infty$  and  $e_1,\ldots ,e_n$  and orthogonal basis of  $(V,\mathfrak{q})$ .
Remark E.14. In the following, we elaborate on the term Clifford group in constraint to previous literature. First, the unconstrained Clifford group is also referred to as the Lipschitz group or Clifford-Lipschitz group in honor of its creator Rudolf Lipschitz [LS09]. Throughout its inventions, several generalizations and versions have been proposed, often varying in details, leading to slightly non-equivalent definitions. For instance, some authors utilize conjugation as an action, others apply the twisted conjugation, while yet another group employs the twisting on the other side of the conjugation operation. Furthermore, some authors require that the elements are homogeneous in the parity grading, where others do not. We settle for a definition that requires homogeneous invertible elements of the Clifford algebra that act via our adjusted twisted conjugation such that elements from the vector space  $V$  land also in  $V$ . The reason for our definition is that we want the adjusted twisted conjugation to act on the whole Clifford algebra  $\operatorname{Cl}(V, \mathfrak{q})$ , not just on the vector space  $V$ . Furthermore, we want it to respect, besides the vector space structure of  $\operatorname{Cl}(V, \mathfrak{q})$ , also the product structure, leading to algebra homomorphisms. In addition, we also want that the action to respect the extended bilinear form  $\mathfrak{h}$ , the orthogonal structure, and the multivector grading. These properties might not (all) be ensured in other definitions with different nuances.
Also note that the name Clifford group might be confused with different groups with the same name in other literature, e.g., with the group of unitary matrices that normalize the Pauli group or with the finite group inside the Clifford algebra that is generated by an orthogonal basis via the geometric product.
Definition E.15. 1. We denote the unconstrained Clifford group of  $\operatorname{Cl}(V, \mathfrak{q})$  as follows:
$$
\tilde {\Gamma} (V, \mathfrak {q}) := \left\{w \in \operatorname {C l} ^ {\times} (V, \mathfrak {q}) \mid \forall v \in V. \rho (w) (v) \in V \right\}. \tag {404}
$$
2. We denote the Clifford group of  $\operatorname{Cl}(V, \mathfrak{q})$  as follows:
$$
\begin{array}{l} \Gamma (V, \mathfrak {q}) := \operatorname {C l} ^ {[ \times ]} (V, \mathfrak {q}) \cap \tilde {\Gamma} (V, \mathfrak {q}) (405) \\ = \left\{w \in \operatorname {C l} ^ {\times} (V, \mathfrak {q}) \mid \eta (w) \in \{\pm 1 \} \wedge \forall v \in V. \rho (w) (v) \in V \right\}. (406) \\ \end{array}
$$
3. We define the special Clifford group as follows:
$$
\Gamma^ {[ 0 ]} (V, \mathfrak {q}) := \tilde {\Gamma} (V, \mathfrak {q}) \cap \mathrm {C l} ^ {[ 0 ]} (V, \mathfrak {q}) = \Gamma (V, \mathfrak {q}) \cap \mathrm {C l} ^ {[ 0 ]} (V, \mathfrak {q}). \tag {407}
$$
Theorem E.16. For  $w \in \Gamma(V, \mathfrak{q})$  and  $x \in \mathrm{Cl}(V, \mathfrak{q})$  we have for all  $m = 0, \ldots, n$ :
$$
\rho (w) \left(x ^ {(m)}\right) = \rho (w) (x) ^ {(m)}. \tag {408}
$$
Proof. We first claim that for  $w \in \Gamma(V, \mathfrak{q})$  the set of elements:
$$
b _ {1} := \rho (w) \left(e _ {1}\right), \dots , b _ {n} := \rho (w) \left(e _ {n}\right), \tag {409}
$$
forms an orthogonal basis of  $(V,\mathfrak{q})$ . Indeed, since  $\rho (w)(e_t)\in V$ , by definition of  $\Gamma (V,\mathfrak{q})$ , the orthogonality relation,  $i\neq j$ :
$$
0 = 2 \mathfrak {b} \left(e _ {i}, e _ {j}\right) = e _ {i} e _ {j} + e _ {j} e _ {i}, \tag {410}
$$
transforms under  $\rho (w)$  to:
$$
\begin{array}{l} 0 = \rho (w) (0) = \rho (w) \left(e _ {i} e _ {j} + e _ {j} e _ {i}\right) (411) \\ = \rho (w) \left(e _ {i}\right) \rho (w) \left(e _ {j}\right) + \rho (w) \left(e _ {j}\right) \rho (w) \left(e _ {i}\right) (412) \\ = b _ {i} b _ {j} + b _ {j} b _ {i} (413) \\ = 2 \mathfrak {b} \left(b _ {i}, b _ {j}\right). (414) \\ \end{array}
$$
This shows that  $b_{1}, \ldots, b_{n}$  is an orthogonal system in  $V$ . Using  $\rho(w^{-1})$  we also see that the system is linear independent and thus an orthogonal basis of  $V$ . By the basis-independence of the multivector grading  $\mathrm{Cl}^{(m)}(V, \mathfrak{q})$ , see Theorem D.27, we then get for:
$$
x = \sum_ {i _ {1} <   \dots <   i _ {m}} c _ {i _ {1}, \dots , i _ {m}} \cdot e _ {i _ {1}} \dots e _ {i _ {m}} \in \mathrm {C l} ^ {(m)} (V, \mathfrak {q}), \tag {415}
$$
the relation:
$$
\rho (w) (x) = \sum_ {i _ {1} <   \dots <   i _ {m}} c _ {i _ {1}, \dots , i _ {m}} \cdot b _ {i _ {1}} \cdot \cdot \cdot b _ {i _ {m}} \in \mathrm {C l} ^ {(m)} (V, \mathfrak {q}). \tag {416}
$$
Corollary E.17. The map:
$$
\rho : \Gamma (V, \mathfrak {q}) \rightarrow \operatorname {A u t} _ {\mathbf {A l g}, \operatorname {g r d}} (\operatorname {C l} (V, \mathfrak {q})) , \quad w \mapsto \rho (w), \tag {417}
$$
is a well-defined group homomorphism from the Clifford group to the group of  $\mathbb{F}$ -algebra automorphisms of  $\operatorname{Cl}(V, \mathfrak{q})$  that preserve the multivector grading of  $\operatorname{Cl}(V, \mathfrak{q})$ . In particular,  $\operatorname{Cl}(V, \mathfrak{q})$  and  $\operatorname{Cl}^{(m)}(V, \mathfrak{q})$  for  $m = 0, \ldots, n$ , are group representations of  $\Gamma(V, \mathfrak{q})$  via  $\rho$ .
Corollary E.18. Let  $F(T_{1},\ldots ,T_{\ell})\in \mathbb{F}[T_{1},\ldots ,T_{\ell}]$  be a polynomial in  $\ell$  variables with coefficients in  $\mathbb{F}$  and let  $k\in \{0,\dots ,n\}$ . Further, consider  $\ell$  elements  $x_{1},\ldots ,x_{\ell}\in \mathrm{Cl}(V,\mathfrak{q})$ . Then for every  $w\in \Gamma (V,\mathfrak{q})$  we get the equivariance property:
$$
\rho (w) \left(F \left(x _ {1},, \dots , x _ {\ell}\right) ^ {(k)}\right) = F (\rho (w) \left(x _ {1}\right), \dots , \rho (w) \left(x _ {\ell}\right)) ^ {(k)}, \tag {418}
$$
where the superscript  $(k)$  indicates the projection onto the multivector grade- $k$ -part of the whole expression.
Example E.19. Let  $w \in V$  with  $\mathfrak{q}(w) \neq 0$ , then  $w \in \Gamma(V, \mathfrak{q})$ .
Proof. It is clear that  $w$  is homogeneous in the parity grading, as  $\eta(w) = -1$ . For  $v \in V$  we get:
$$
\rho (w) (v) = r _ {w} (v) = v - 2 \frac {\mathfrak {b} (v , w)}{\mathfrak {q} (w)} w \in V. \tag {419}
$$
This shows  $w\in \Gamma (V,\mathfrak{q})$
Example E.20. Let  $e, f \in V$  with  $\mathfrak{b}(v, f) = 0$  for all  $v \in V$ , and put:  $\gamma := 1 + ef \in \mathrm{Cl}^{[0]}(V, \mathfrak{q})$ . Then  $\gamma \in \Gamma(V, \mathfrak{q})$ .
Proof. First, note that for all  $v \in V$  we get:
$$
f v = - v f + 2 \mathfrak {b} (f, v) = - v f. \tag {420}
$$
Next, we see that:
$$
\gamma^ {- 1} = 1 - e f. \tag {421}
$$
Indeed, we get:
$$
\begin{array}{l} (1 + e f) (1 - e f) = 1 + e f - e f - e f e f (422) \\ = 1 + e f f e (423) \\ = 1 + \underbrace {\mathfrak {q} (f)} _ {= 0} \mathfrak {q} (e) (424) \\ = 1. (425) \\ \end{array}
$$
Now consider  $v \in V$ . We then have:
$$
\begin{array}{l} \rho (\gamma) (v) = \alpha (\gamma) v \gamma^ {- 1} (426) \\ = (1 + e f) v (1 - e f) (427) \\ = (1 + e f) (v - v e f) (428) \\ = v + e f v - v e f - e f v e f (429) \\ = v - e v f - v e f - e v e f f (430) \\ = v - (e v + v e) f - \underbrace {\mathfrak {q} (f)} _ {= 0} e v e (431) \\ = v - 2 \mathrm {b} (e, v) f (432) \\ \in V. (433) \\ \end{array}
$$
This shows  $\gamma \in \Gamma (V,\mathfrak{q})$
Example E.21. Let  $f_1, \ldots, f_r \in V$  be a basis of the radical subspace  $\mathcal{R}$  of  $(V, \mathfrak{q})$ . In particular, we have  $\mathfrak{b}(v, f_j) = 0$  for all  $v \in V$ . Then we put:
$$
g := 1 + h, \quad \text {w i t h} \quad h \quad \in \operatorname {s p a n} \left\{f _ {k _ {1}} \dots f _ {k _ {s}} \mid s \geq 2 \underline {{e v e n}}, 1 \leq k _ {1} <   k _ {2} <   \dots k _ {s} \leq r \right\} \subseteq \operatorname {C l} (V, \mathfrak {q}). \tag {434}
$$
Then we claim that  $g\in \Gamma (V,\mathfrak{q})$  and  $\rho (g) = \mathrm{id}_{\operatorname {Cl}(V,\mathfrak{q})}$
Proof. Since we restrict to even products it is clear that  $g \in \mathrm{Cl}^{[0]}(V, \mathfrak{q})$ . Furthermore, note that, since  $h$  lives in the radical subalgebra, there exists a number  $k \geq 1$  such that:
$$
h ^ {k + 1} = 0. \tag {435}
$$
Then we get that:
$$
g ^ {- 1} = 1 - h + h ^ {2} + \dots + (- 1) ^ {k} h ^ {k}. \tag {436}
$$
$$
\begin{array}{l} (1 + h) \left(\sum_ {l = 0} ^ {k} (- 1) ^ {l} h ^ {l}\right) = \sum_ {l = 0} ^ {k} (- 1) ^ {l} h ^ {l} + \sum_ {l = 0} ^ {k} (- 1) ^ {l} h ^ {l + 1} (437) \\ = 1 + \sum_ {l = 1} ^ {k} (- 1) ^ {l} h ^ {l} - \sum_ {l = 1} ^ {k} (- 1) ^ {l} h ^ {l} + (- 1) ^ {k} \underbrace {h ^ {k + 1}} _ {= 0} (438) \\ = 1. (439) \\ \end{array}
$$
Furthermore,  $g$  lies in the center  $\mathfrak{Z}(\operatorname{Cl}(V, \mathfrak{q}))$  of  $\operatorname{Cl}(V, \mathfrak{q})$  by Theorem D.47. Then for  $v \in V$  we get:
$$
\begin{array}{l} \rho (g) (v) = \alpha (g) v g ^ {- 1} (440) \\ = g v g ^ {- 1} (441) \\ = v g g ^ {- 1} (442) \\ = v (443) \\ \in V. (444) \\ \end{array}
$$
So,  $g \in \Gamma(V, \mathfrak{q})$  and acts as the identity on  $V$ , and thus on  $\operatorname{Cl}(V, \mathfrak{q})$ , via  $\rho$ .
E.3 The Structure of the Clifford Group
We have identified the Clifford group and its action on the algebra. In particular, our adjusted twisted conjugation preserves the parity and multivector grading, and reduces to a reflection when restricted to  $V$ . We now want to further investigate how the Clifford and the Clifford groups act via the twisted conjugation  $\rho$  on  $V$  and  $\operatorname{Cl}(V, \mathfrak{q})$ . We again denote by  $\mathcal{R} \subseteq V$  the radical subspace of  $V$  w.r.t.  $\mathfrak{q}$ . We follow and extend the analysis of [Cru80, Cru90, DKL10].
We first investigate the kernel of the twisted action.
Corollary E.22 (The kernel of the twisted conjugation). the twisted conjugation:
1. We have the following identity for
$$
\ker \left(\rho | _ {\mathrm {C l} ^ {\times} (V, \mathfrak {q})}\right) := \left\{w \in \mathrm {C l} ^ {\times} (V, \mathfrak {q}) \mid \rho (w) = \operatorname {i d} _ {\mathrm {C l} (V, \mathfrak {q})} \right\} \stackrel {!} {=} \bigwedge^ {\times} (\mathcal {R}). \tag {445}
$$
2. For the twisted conjugation restricted to the unconstrained Clifford, group and to  $V$  the kernel is given by:
$$
\ker \left(\rho : \tilde {\Gamma} (V, \mathfrak {q}) \rightarrow \operatorname {G L} (V)\right) := \left\{w \in \tilde {\Gamma} (V, \mathfrak {q}) \mid \rho (w) | _ {V} = \mathrm {i d} _ {V} \right\} \overset {!} {=} \bigwedge^ {\times} (\mathcal {R}). \tag {446}
$$
3. For the twisted conjugation restricted to the Clifford group, the kernel is given by:
$$
\begin{array}{l} \ker \left(\rho : \Gamma (V, \mathfrak {q}) \rightarrow \operatorname {A u t} _ {\mathbf {A l g}} (\operatorname {C l} (V, \mathfrak {q}))\right) := \left\{w \in \Gamma (V, \mathfrak {q}) \mid \rho (w) = \operatorname {i d} _ {\operatorname {C l} (V, \mathfrak {q})} \right\} (447) \\ \stackrel {!} {=} \bigwedge^ {[ \times ]} (\mathcal {R}). (448) \\ \end{array}
$$
Proof. This follows directly from the characterizing of the twisted center of  $\operatorname{Cl}(V, \mathfrak{q})$  by Theorem D.50. Note that we have for  $w \in \operatorname{Cl}^{\times}(V, \mathfrak{q})$ :
$$
\begin{array}{l} \rho (w) = \operatorname {i d} _ {\operatorname {C l} (V, \mathfrak {q})} \Longleftrightarrow \forall x \in \operatorname {C l} (V, \mathfrak {q}). \rho (w) (x) = x (449) \\ \Longleftrightarrow \forall x \in \operatorname {C l} (V, \mathfrak {q}). w x ^ {[ 0 ]} w ^ {- 1} + \alpha (w) x ^ {[ 1 ]} w ^ {- 1} = x (450) \\ \Longleftrightarrow \forall x \in \operatorname {C l} (V, \mathfrak {q}). w x ^ {[ 0 ]} + \alpha (w) x ^ {[ 1 ]} = x w (451) \\ \Longleftrightarrow w \in \Re (\operatorname {C l} (V, \mathfrak {q})) = \bigwedge (\mathcal {R}). (452) \\ \end{array}
$$
From this follows that:
$$
w \in \mathrm {C l} ^ {\times} (V, \mathfrak {q}) \cap \bigwedge (\mathcal {R}) = \bigwedge^ {\times} (\mathcal {R}). \tag {453}
$$
The other points follow similarly with Theorem D.50.
For the last point also note:
$$
\Gamma (V, \mathfrak {q}) = \tilde {\Gamma} (V, \mathfrak {q}) \cap \operatorname {C l} ^ {[ \times ]} (V, \mathfrak {q}), \quad \bigwedge^ {\times} (\mathcal {R}) \cap \operatorname {C l} ^ {[ \times ]} (V, \mathfrak {q}) = \bigwedge^ {[ \times ]} (\mathcal {R}). \tag {454}
$$
This shows the claims.
Lemma E.23. 1. For every  $w\in \mathrm{Cl}^{\times}(V,\mathfrak{q})$  we have:
$$
\rho (w) | _ {\mathcal {R}} = \mathrm {i d} _ {\mathcal {R}}. \tag {455}
$$
2. For every  $w\in \mathrm{Cl}^{[\times ]}(V,\mathfrak{q})$  we have:
$$
\rho (w) | _ {\Lambda (\mathcal {R})} = \operatorname {i d} _ {\Lambda (\mathcal {R})}. \tag {456}
$$
3. For every  $g \in \bigwedge^{\times}(\mathcal{R})$  we have:
$$
\rho (g) | _ {V} = \operatorname {i d} _ {V}. \tag {457}
$$
4. For every  $g\in \bigwedge^{[\times ]}(\mathcal{R})$  we have:
$$
\rho (g) | _ {\operatorname {C l} (V, \mathfrak {q})} = \mathrm {i d} _ {\operatorname {C l} (V, \mathfrak {q})}. \tag {458}
$$
Proof. This directly follows from the twisted commutation relationship, see Lemma D.42. For  $w \in \mathrm{Cl}^{\times}(V, \mathfrak{q})$  and  $f \in \mathcal{R} \subseteq V$  we have:
$$
\rho (w) (f) = \alpha (w) f w ^ {- 1} = f w w ^ {- 1} = f. \tag {459}
$$
For  $w \in \mathrm{Cl}^{[\times]}(V, \mathfrak{q})$  the map  $\rho(w)$  is an algebra automorphism of  $\operatorname{Cl}(V, \mathfrak{q})$  and satisfies  $\rho(w)(f) = f$  for all  $f \in \mathcal{R}$ . So for every  $y \in \bigwedge(\mathcal{R})$  we have:
$$
\begin{array}{l} \rho (w) (y) = \rho (w) \left(\sum_ {i \in I} c _ {i} \cdot f _ {k _ {i}} \dots f _ {l _ {i}}\right) (460) \\ = \sum_ {i \in I} c _ {i} \cdot \rho (w) \left(f _ {k _ {i}}\right) \dots \rho (w) \left(f _ {l _ {i}}\right) (461) \\ = \sum_ {i \in I} c _ {i} \cdot f _ {k _ {i}} \dots f _ {l _ {i}} (462) \\ = y. (463) \\ \end{array}
$$
For  $g\in \bigwedge^{\times}(\mathcal{R})$  and  $v\in V$  we get:
$$
\rho (g) (v) = \alpha (g) v g ^ {- 1} = v g g ^ {- 1} = v. \tag {464}
$$
For  $g \in \bigwedge^{[\times]}(\mathcal{R})$  the map  $\rho(g)$  is an algebra automorphism of  $\operatorname{Cl}(V, \mathfrak{q})$  and satisfies  $\rho(g)(v) = v$  for all  $v \in V$ . As above we see that for  $x \in \operatorname{Cl}(V, q)$  we get:
$$
\begin{array}{l} \rho (g) (x) = \rho (g) \left(\sum_ {i \in I} c _ {i} \cdot v _ {k _ {i}} \dots v _ {l _ {i}}\right) (465) \\ = \sum_ {i \in I} c _ {i} \cdot \rho (g) \left(v _ {k _ {i}}\right) \dots \rho (g) \left(v _ {l _ {i}}\right) (466) \\ = \sum_ {i \in I} c _ {i} \cdot v _ {k _ {i}} \dots v _ {l _ {i}} (467) \\ = x. (468) \\ \end{array}
$$
This shows all the claims.
From Lemma E.23 we see that  $\rho(w)|_{\Lambda(\mathcal{R})} = \mathrm{id}_{\Lambda(\mathcal{R})}$  for  $w \in \operatorname{Cl}^{[\times]}(V, \mathfrak{q})$ . Together with Corollary E.18 we arrive at a slightly more general version that allows one to parameterize polynomials not just with coefficients from  $\mathbb{F}$ , but also with elements from  $\Lambda(\mathcal{R})$ , and still get the equivariance w.r.t. the Clifford group  $\Gamma(V, \mathfrak{q})$ :
Corollary E.24. 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  $w\in \Gamma (V,\mathfrak{q})$  we get the equivariance property:
$$
\rho (w) \left(F \left(x _ {1},, \dots , x _ {\ell}, y _ {1}, \dots , y _ {s}\right) ^ {(k)}\right) = F (\rho (w) \left(x _ {1}\right), \dots , \rho (w) \left(x _ {\ell}\right), y _ {1}, \dots , y _ {s}) ^ {(k)}, \tag {469}
$$
We now investigate the image/range of the twisted conjugation.
Theorem E.25 (The range of the twisted conjugation). The image/range of the Clifford group  $\Gamma(V, \mathfrak{q})$  under the twisted conjugation restricted to  $V$  coincides with all orthogonal automorphisms of  $(V, \mathfrak{q})$  that restrict to the identity  $\mathrm{id}_{\mathcal{R}}$  of the radical subspace  $\mathcal{R} \subseteq V$  of  $(V, \mathfrak{q})$ :
$$
\operatorname {r a n} (\rho : \Gamma (V, \mathfrak {q}) \rightarrow \mathrm {G L} (V)) = \mathrm {O} _ {\mathcal {R}} (V, \mathfrak {q}). \tag {470}
$$
Again, recall that the kernel is given by:
$$
\ker (\rho : \Gamma (V, \mathfrak {q}) \rightarrow \operatorname {G L} (V)) = \bigwedge^ {[ \times ]} (\mathcal {R}). \tag {471}
$$
Proof. We first show that the range of  $\rho$  when restricted to  $V$  will constitute an orthogonal automorphism of  $(V, \mathfrak{q})$ . For this let  $e_1, \ldots, e_n$  be an orthogonal basis of  $(V, \mathfrak{q})$  and  $\mathcal{R}$  the radical subspace of  $(V, \mathfrak{q})$ ,  $r = \dim \mathcal{R}$ . W.l.o.g. we can assume that  $e_1, \ldots, e_m, e_{m+1}, \ldots, e_{m+r}$  with:
$$
\mathcal {R} = \operatorname {s p a n} \left\{e _ {m + 1}, \dots , e _ {m + r} \right\}, \quad E := \operatorname {s p a n} \left\{e _ {1}, \dots , e _ {m} \right\}. \tag {472}
$$