SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Further, the Clifford algebra  $\operatorname{Cl}(V, \mathfrak{q})$  is fully characterized by the following property.
Theorem D.4 (The universal property of the Clifford algebra). For every  $\mathbb{F}$ -algebra $^{10}$  (an algebra over field  $\mathbb{F}$ )  $\mathcal{A}$  and every  $\mathbb{F}$ -linear map  $f: V \to \mathcal{A}$  such that for all  $v \in V$  we have:
$$
f (v) ^ {2} = \mathfrak {q} (v) \cdot 1 _ {\mathcal {A}}, \tag {53}
$$
there exists a unique  $\mathbb{F}$ -algebra homomorphism<sup>11</sup>  $\bar{f}:\operatorname {Cl}(V,\mathfrak{q})\to \mathcal{A}$  such that  $\bar{f} (v) = f(v)$  for all  $v\in V$
More explicitly, if  $f$  satisfies equation 53 and  $x \in \operatorname{Cl}(V, \mathfrak{q})$ . Then we can take any representation of  $x$  of the following form:
$$
x = c _ {0} + \sum_ {i \in I} c _ {i} \cdot v _ {i, 1} \dots v _ {i, k _ {i}}, \tag {54}
$$
with finite index sets  $I$  and  $k_{i} \in \mathbb{N}$  and coefficients  $c_{0}, c_{i} \in \mathbb{F}$  and vectors  $v_{i,j} \in V$ ,  $j = 1, \ldots, k_{i}$ ,  $i \in I$ , and, then we can compute  $\bar{f}(x)$  by the following formula (without ambiguity):
$$
\bar {f} (x) = c _ {0} \cdot 1 _ {\mathcal {A}} + \sum_ {i \in I} c _ {i} \cdot f \left(v _ {i, 1}\right) \dots f \left(v _ {i, k _ {i}}\right). \tag {55}
$$
In the following, we will often denote  $\bar{f}$  again with  $f$  without further indication.
D.2 The Multivector Filtration and the Grade
Note that the Clifford algebra is a filtered algebra.
Definition D.5. We define the multivector filtration $^{12}$  of  $\mathrm{Cl}(V, \mathfrak{q})$  for grade  $m \in \mathbb{N}_0$  as follows:
$$
\operatorname {C l} ^ {(\leq m)} (V, \mathfrak {q}) := \pi \left(\mathrm {T} ^ {(\leq m)} (V)\right), \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \left. T ^ {(\leq m)} (V) := \bigoplus_ {l = 0} ^ {m} V ^ {\otimes l}. \right. \tag {56}
$$
Remark D.6. Note that we really get a filtration on the space  $\mathrm{Cl}(V,\mathfrak{q})$ :
$$
\mathbb {F} = \operatorname {C l} ^ {(\leq 0)} (V, \mathfrak {q}) \subseteq \operatorname {C l} ^ {(\leq 1)} (V, \mathfrak {q}) \subseteq \operatorname {C l} ^ {(\leq 2)} (V, \mathfrak {q}) \subseteq \dots \subseteq \operatorname {C l} ^ {(\leq n)} (V, \mathfrak {q}) = \operatorname {C l} (V, \mathfrak {q}). \tag {57}
$$
Furthermore, note that this is compatible with the algebra structure of  $\operatorname{Cl}(V, \mathfrak{q})$ , i.e. for  $i, j \geq 0$  we get:
$$
x \in \operatorname {C l} ^ {(\leq i)} (V, \mathfrak {q}) \quad \wedge \quad y \in \operatorname {C l} ^ {(\leq j)} (V, \mathfrak {q}) \quad \Longrightarrow \quad x y \in \operatorname {C l} ^ {(\leq i + j)} (V, \mathfrak {q}). \tag {58}
$$
Together with the equality:  $\pi (\mathrm{T}^{(\leq m)}(V)) = \mathrm{Cl}^{(\leq m)}(V,\mathfrak{q})$  for all  $m$ , we see that the natural map:
$$
\pi : \mathrm {T} (V) \rightarrow \operatorname {C l} (V, \mathfrak {q}), \tag {59}
$$
is a surjective homomorphism of filtered algebras. Indeed, since  $\operatorname{Cl}(V, \mathfrak{q})$  is a well-defined algebra, since we modded out a two-sided ideal,  $\pi$  is clearly a homomorphism of filtered algebras. As a quotient map  $\pi$  is automatically surjective.
Definition D.7 (The grade of an element). For  $x \in \operatorname{Cl}(V, \mathfrak{q}) \setminus \{0\}$  we define its grade through the following condition:
$$
\operatorname {g r d} x := k \quad \text {s u c h t h a t} \quad x \in \operatorname {C l} ^ {(\leq k)} (V, \mathfrak {q}) \backslash \operatorname {C l} ^ {(\leq k - 1)} (V, \mathfrak {q}), \tag {60}
$$
$$
\operatorname {g r d} 0 := - \infty . \tag {61}
$$
<sup>10</sup>For the purpose of this text an algebra is always considered to be associative and unital (containing an identity element), but not necessarily commutative.
An algebra homomorphism is both linear and multiplicative.
$^{12}$ A filtration  $\mathcal{F}$  is an indexed family  $(A_{i})_{i\in I}$  ( $I$  is an ordered index set) of subsets of an algebraic structure  $A$  such that for  $i\leq j:A_i\subseteq A_j$ .
D.3 The Parity Grading
In this section, we introduce the parity grading of the Clifford algebra. We will use the parity grading to later construct the (adjusted) twisted conjugation map, which will be used as a group action on the Clifford algebra.
Definition D.8 (The main involution). The linear map:
$$
\alpha : V \rightarrow \operatorname {C l} (V, \mathfrak {q}), \quad \alpha (v) := - v, \tag {62}
$$
satisfies  $(-v)^2 = v^2 = \mathfrak{q}(v)$ . The universal property of  $\operatorname{Cl}(V, \mathfrak{q})$  thus extends  $\alpha$  to a unique algebra homomorphism:
$$
\begin{array}{l} \alpha : \operatorname {C l} (V, \mathfrak {q}) \rightarrow \operatorname {C l} (V, \mathfrak {q}), \quad \alpha \left(c _ {0} + \sum_ {i \in I} c _ {i} \cdot v _ {i, 1} \dots v _ {i, k _ {i}}\right) (63) \\ = c _ {0} + \sum_ {i \in I} c _ {i} \cdot \alpha \left(v _ {i, 1}\right) \dots \alpha \left(v _ {i, k _ {i}}\right) (64) \\ = c _ {0} + \sum_ {i \in I} (- 1) ^ {k _ {i}} \cdot c _ {i} \cdot v _ {i, 1} \dots v _ {i, k _ {i}}, (65) \\ \end{array}
$$
for any finite sum representation with  $v_{i,j} \in V$  and  $c_{i} \in \mathbb{F}$ . This extension  $\alpha$  will be called the main involution<sup>13</sup> of  $\operatorname{Cl}(V, \mathfrak{q})$ .
Definition D.9 (Parity grading). The main involution  $\alpha$  of  $\operatorname{Cl}(V, \mathfrak{q})$  now defines the parity grading of  $\operatorname{Cl}(V, \mathfrak{q})$  via the following homogeneous parts:
$$
\operatorname {C l} ^ {[ 0 ]} (V, \mathfrak {q}) := \{x \in \operatorname {C l} (V, \mathfrak {q}) \mid \alpha (x) = x \}, \tag {66}
$$
$$
\operatorname {C l} ^ {[ 1 ]} (V, \mathfrak {q}) := \{x \in \operatorname {C l} (V, \mathfrak {q}) \mid \alpha (x) = - x \}. \tag {67}
$$
With this we get the direct sum decomposition:
$$
\operatorname {C l} (V, \mathfrak {q}) = \operatorname {C l} ^ {[ 0 ]} (V, \mathfrak {q}) \oplus \operatorname {C l} ^ {[ 1 ]} (V, \mathfrak {q}), \tag {68}
$$
$$
x = x ^ {[ 0 ]} + x ^ {[ 1 ]}, \quad x ^ {[ 0 ]} := \frac {1}{2} (x + \alpha (x)), \quad x ^ {[ 1 ]} := \frac {1}{2} (x - \alpha (x)), \tag {69}
$$
with the homogeneous parts  $x^{[0]}\in \mathrm{Cl}^{[0]}(V,\mathfrak{q})$  and  $x^{[1]}\in \mathrm{Cl}^{[1]}(V,\mathfrak{q})$ .
We define the parity of an (homogeneous) element  $x \in \operatorname{Cl}(V, \mathfrak{q})$  as follows:
$$
\operatorname {p r t} (x) := \left\{ \begin{array}{l l} 0 & \text {i f} x \in \operatorname {C l} ^ {[ 0 ]} (V, \mathfrak {q}), \\ 1 & \text {i f} x \in \operatorname {C l} ^ {[ 1 ]} (V, \mathfrak {q}). \end{array} \right. \tag {70}
$$
Definition D.10 ( $\mathbb{Z}/2\mathbb{Z}$ -graded algebras). An  $\mathbb{F}$ -algebra  $(\mathcal{A}, +, \cdot)$  together with a direct sum decomposition of sub-vector spaces:
$$
\mathcal {A} = \mathcal {A} ^ {[ 0 ]} \oplus \mathcal {A} ^ {[ 1 ]}, \tag {71}
$$
is called a  $\mathbb{Z} / 2\mathbb{Z}$ -graded algebra if for every  $i, j \in \mathbb{Z} / 2\mathbb{Z}$  we always have:
$$
x \in \mathcal {A} ^ {[ i ]} \quad \wedge \quad y \in \mathcal {A} ^ {[ j ]} \quad \Longrightarrow \quad x \cdot y \in \mathcal {A} ^ {[ i + j ]}. \tag {72}
$$
Note that  $[i + j]$  is meant here to be computed modulo 2. The  $\mathbb{Z} / 2\mathbb{Z}$ -grade of an (homogeneous) element  $x \in \mathcal{A}$  will also be called the parity of  $x$ :
$$
\operatorname {p r t} (x) := \left\{ \begin{array}{l l} 0 & \text {i f} x \in \mathcal {A} ^ {[ 0 ]}, \\ 1 & \text {i f} x \in \mathcal {A} ^ {[ 1 ]}. \end{array} \right. \tag {73}
$$
The above requirement then implies for homogeneous elements  $x, y \in \mathcal{A}$  the relation:
$$
\operatorname {p r t} (x \cdot y) = \operatorname {p r t} (x) + \operatorname {p r t} (y) \mod 2. \tag {74}
$$
We can now summarize the results of this section as follows:
Theorem D.11. The Clifford algebra  $\operatorname{Cl}(V, \mathfrak{q})$  is a  $\mathbb{Z}/2\mathbb{Z}$ -graded algebra in its parity grading.
An involution is a map that is its own inverse.
D.4 The Dimension of the Clifford Algebra
In this subsection we determine the dimension of the Clifford algebra, which allows us to construct bases for the Clifford algebra.
In the following, we again let  $\mathbb{F}$  be any field of  $\mathrm{char}(\mathbb{F})\neq 2$
Definition/Lemma D.12 (The twisted tensor product of  $\mathbb{Z}/2\mathbb{Z}$ -graded  $\mathbb{F}$ -algebras). Let  $\mathcal{A}$  and  $\mathcal{B}$  be two  $\mathbb{Z}/2\mathbb{Z}$ -graded algebras over  $\mathbb{F}$ . Then their twisted tensor product  $\mathcal{A} \hat{\otimes} \mathcal{B}$  is defined via the usual tensor product  $\mathcal{A} \otimes \mathcal{B}$  of  $\mathbb{F}$ -vector spaces, but where the product is defined on homogeneous elements  $a_1, a_2 \in \mathcal{A}, b_1, b_2 \in \mathcal{B}$  via:
$$
\left(a _ {1} \hat {\otimes} b _ {1}\right) \cdot \left(a _ {2} \hat {\otimes} b _ {2}\right) := (- 1) ^ {\operatorname {p r t} \left(b _ {1}\right) \cdot \operatorname {p r t} \left(a _ {2}\right)} \left(a _ {1} a _ {2}\right) \hat {\otimes} \left(b _ {1} b _ {2}\right). \tag {75}
$$
This turns  $\mathcal{A}\hat{\otimes}\mathcal{B}$  also into a  $\mathbb{Z} / 2\mathbb{Z}$ -graded algebra over  $\mathbb{F}$  with the  $\mathbb{Z} / 2\mathbb{Z}$ -grading:
$$
\left(\mathcal {A} \hat {\otimes} \mathcal {B}\right) ^ {[ 0 ]} := \left(\mathcal {A} ^ {[ 0 ]} \otimes \mathcal {B} ^ {[ 0 ]}\right) \oplus \left(\mathcal {A} ^ {[ 1 ]} \otimes \mathcal {B} ^ {[ 1 ]}\right), \tag {76}
$$
$$
\left(\mathcal {A} \hat {\otimes} \mathcal {B}\right) ^ {[ 1 ]} := \left(\mathcal {A} ^ {[ 0 ]} \otimes \mathcal {B} ^ {[ 1 ]}\right) \oplus \left(\mathcal {A} ^ {[ 1 ]} \otimes \mathcal {B} ^ {[ 0 ]}\right). \tag {77}
$$
In particular, if  $a \in \mathcal{A}$ ,  $b \in \mathcal{B}$  are homogeneous elements then we have:
$$
\operatorname {p r t} _ {\mathcal {A} \hat {\otimes} \mathcal {B}} (a \hat {\otimes} b) = \operatorname {p r t} _ {\mathcal {A}} (a) + \operatorname {p r t} _ {\mathcal {B}} (b) \mod 2. \tag {78}
$$
Proof. By definition we already know that  $\mathcal{A}\hat{\otimes}\mathcal{B}$  is an  $\mathbb{F}$ -vector space. So we only need to investigate the multiplication and  $\mathbb{Z}/2\mathbb{Z}$ -grading.
For homogenous elements  $a_1, a_2, a_3 \in \mathcal{A}, b_1, b_2, b_3 \in \mathcal{B}$  we have:
$$
\begin{array}{l} \left(\left(a _ {1} \hat {\otimes} b _ {1}\right) \cdot \left(a _ {2} \hat {\otimes} b _ {2}\right)\right) \cdot \left(a _ {3} \hat {\otimes} b _ {3}\right) (79) \\ = (- 1) ^ {\operatorname {p r t} \left(b _ {1}\right) \cdot \operatorname {p r t} \left(a _ {2}\right)} \left(\left(a _ {1} a _ {2}\right) \hat {\otimes} \left(b _ {1} b _ {2}\right)\right) \cdot \left(a _ {3} \hat {\otimes} b _ {3}\right) (80) \\ = (- 1) ^ {\operatorname {p r t} \left(b _ {1}\right) \cdot \operatorname {p r t} \left(a _ {2}\right)} \cdot (- 1) ^ {\operatorname {p r t} \left(b _ {1} b _ {2}\right) \cdot \operatorname {p r t} \left(a _ {3}\right)} \left(a _ {1} a _ {2} a _ {3}\right) \hat {\otimes} \left(b _ {1} b _ {2} b _ {3}\right) (81) \\ = (- 1) ^ {\operatorname {p r t} \left(b _ {1}\right) \cdot \operatorname {p r t} \left(a _ {2}\right) + \operatorname {p r t} \left(b _ {1}\right) \cdot \operatorname {p r t} \left(a _ {3}\right) + \operatorname {p r t} \left(b _ {2}\right) \cdot \operatorname {p r t} \left(a _ {3}\right)} \left(a _ {1} a _ {2} a _ {3}\right) \hat {\otimes} \left(b _ {1} b _ {2} b _ {3}\right) (82) \\ = (- 1) ^ {\operatorname {p r t} \left(b _ {1}\right) \cdot \operatorname {p r t} \left(a _ {2} a _ {3}\right)} \cdot (- 1) ^ {\operatorname {p r t} \left(b _ {2}\right) \cdot \operatorname {p r t} \left(a _ {3}\right)} \left(a _ {1} a _ {2} a _ {3}\right) \hat {\otimes} \left(b _ {1} b _ {2} b _ {3}\right) (83) \\ = (- 1) ^ {\operatorname {p r t} \left(b _ {2}\right) \cdot \operatorname {p r t} \left(a _ {3}\right)} \left(a _ {1} \hat {\otimes} b _ {1}\right) \cdot \left(\left(a _ {2} a _ {3}\right) \hat {\otimes} \left(b _ {2} b _ {3}\right)\right) (84) \\ = \left(a _ {1} \hat {\otimes} b _ {1}\right) \cdot \left(\left(a _ {2} \hat {\otimes} b _ {2}\right) \cdot \left(a _ {3} \hat {\otimes} b _ {3}\right)\right). (85) \\ \end{array}
$$
This shows associativity of multiplication on homogeneous elements, which extends by linearity to general elements. The distributive law is clear.
To check that we have a  $\mathbb{Z} / 2\mathbb{Z}$ -grading, note that for homogeneous elements  $a_1, a_2 \in \mathcal{A}, b_1, b_2 \in \mathcal{B}$  we have:
$$
\begin{array}{l} \operatorname {p r t} _ {\mathcal {A} \hat {\otimes} \mathcal {B}} \left(\left(a _ {1} \hat {\otimes} b _ {1}\right) \cdot \left(a _ {2} \hat {\otimes} b _ {2}\right)\right) = \operatorname {p r t} _ {\mathcal {A} \hat {\otimes} \mathcal {B}} \left(\left((- 1) ^ {\operatorname {p r t} \left(a _ {2}\right) \cdot \operatorname {p r t} \left(b _ {1}\right)} a _ {1} a _ {2}\right) \hat {\otimes} \left(b _ {1} b _ {2}\right)\right) (86) \\ = \operatorname {p r t} _ {\mathcal {A}} \left(a _ {1} a _ {2}\right) + \operatorname {p r t} _ {\mathcal {B}} \left(b _ {1} b _ {2}\right) (87) \\ = \operatorname {p r t} _ {\mathcal {A}} \left(a _ {1}\right) + \operatorname {p r t} _ {\mathcal {A}} \left(a _ {2}\right) + \operatorname {p r t} _ {\mathcal {B}} \left(b _ {1}\right) + \operatorname {p r t} _ {\mathcal {B}} \left(b _ {2}\right) (88) \\ = \operatorname {p r t} _ {\mathcal {A} \hat {\otimes} \mathcal {B}} \left(a _ {1} \hat {\otimes} b _ {1}\right) + \operatorname {p r t} _ {\mathcal {A} \hat {\otimes} \mathcal {B}} \left(a _ {2} \hat {\otimes} b _ {2}\right) \quad \mathrm {m o d} 2. (89) \\ \end{array}
$$
The general case follows by linear combinations.
Remark D.13 (The universal property of the twisted tensor product of  $\mathbb{Z}/2\mathbb{Z}$ -graded  $\mathbb{F}$ -algebras). Let  $\mathcal{A}_1, \mathcal{A}_2, \mathcal{B}$  be  $\mathbb{Z}/2\mathbb{Z}$ -graded  $\mathbb{F}$ -algebras. Consider  $\mathbb{Z}/2\mathbb{Z}$ -graded  $\mathbb{F}$ -algebra homomorphisms:
$$
\psi_ {1}: \mathcal {A} _ {1} \rightarrow \mathcal {B}, \quad \psi_ {2}: \mathcal {A} _ {2} \rightarrow \mathcal {B}, \tag {90}
$$
such that for all homogeneous elements  $a_1 \in \mathcal{A}_1$  and  $a_2 \in \mathcal{A}_2$  we have:
$$
\psi_ {1} \left(a _ {1}\right) \cdot \psi_ {2} \left(a _ {2}\right) = (- 1) ^ {\operatorname {p r t} \left(a _ {1}\right) \cdot \operatorname {p r t} \left(a _ {2}\right)} \cdot \psi_ {2} \left(a _ {2}\right) \cdot \psi_ {1} \left(a _ {1}\right). \tag {91}
$$
Then there exists a unique  $\mathbb{Z} / 2\mathbb{Z}$ -graded  $\mathbb{F}$ -algebra homomorphism:
$$
\psi : \mathcal {A} _ {1} \hat {\otimes} \mathcal {A} _ {2} \rightarrow \mathcal {B}, \tag {92}
$$
$$
\psi \circ \phi_ {1} = \psi_ {1}, \quad \psi \circ \phi_ {2} = \psi_ {2}, \tag {93}
$$
where  $\phi_1, \phi_2$  are the following  $\mathbb{Z}/2\mathbb{Z}$ -graded  $\mathbb{F}$ -algebra homomorphisms:
$$
\phi_ {1}: \mathcal {A} _ {1} \rightarrow \mathcal {A} _ {1} \hat {\otimes} \mathcal {A} _ {2}, \quad a _ {1} \mapsto a _ {1} \hat {\otimes} 1, \tag {94}
$$
$$
\phi_ {2}: \mathcal {A} _ {2} \rightarrow \mathcal {A} _ {1} \hat {\otimes} \mathcal {A} _ {2}, \quad a _ {2} \mapsto 1 \hat {\otimes} a _ {2}, \tag {95}
$$
which satisfy for all homogeneous elements  $a_1 \in \mathcal{A}_1$  and  $a_2 \in \mathcal{A}_2$ :
$$
\phi_ {1} \left(a _ {1}\right) \cdot \phi_ {2} \left(a _ {2}\right) = (- 1) ^ {\operatorname {p r t} \left(a _ {1}\right) \cdot \operatorname {p r t} \left(a _ {2}\right)} \cdot \phi_ {2} \left(a _ {2}\right) \cdot \phi_ {1} \left(a _ {1}\right). \tag {96}
$$
Furthermore,  $\mathcal{A}_1\hat{\otimes}\mathcal{A}_2$  together with  $\phi_1,\phi_2$  is uniquely characterized as a  $\mathbb{Z} / 2\mathbb{Z}$ -graded  $\mathbb{F}$ -algebra by the above property (when considering all possible such  $\mathcal{B}$  and  $\psi_{1},\psi_{2}$ ).
Proposition D.14. Let  $(V, \mathfrak{q})$  be a finite dimensional quadratic vector space over  $\mathbb{F}$ ,  $\mathrm{char}(\mathbb{F}) \neq 2$ , with an orthogonal sum decomposition:
$$
(V, \mathfrak {q}) = \left(V _ {1}, \mathfrak {q} _ {1}\right) \oplus \left(V _ {2}, \mathfrak {q} _ {2}\right). \tag {97}
$$
Then the inclusion maps:  $\operatorname{Cl}(V_i, \mathfrak{q}_i) \to \operatorname{Cl}(V, \mathfrak{q})$  induce an isomorphism of  $\mathbb{Z}/2\mathbb{Z}$ -graded  $\mathbb{F}$ -algebras:
$$
\operatorname {C l} (V, \mathfrak {q}) \cong \operatorname {C l} \left(V _ {1}, \mathfrak {q} _ {1}\right) \hat {\otimes} \operatorname {C l} \left(V _ {2}, \mathfrak {q} _ {2}\right). \tag {98}
$$
$$
\dim \operatorname {C l} (V, \mathfrak {q}) = \dim \operatorname {C l} \left(V _ {1}, \mathfrak {q} _ {1}\right) \cdot \dim \operatorname {C l} \left(V _ {2}, \mathfrak {q} _ {2}\right). \tag {99}
$$
Proof. First consider the canonical inclusion maps,  $l = 1,2$ :
$$
\phi_ {l}: \operatorname {C l} \left(V _ {l}, \mathfrak {q} _ {l}\right)\rightarrow \operatorname {C l} (V, \mathfrak {q}), \quad x _ {l} \mapsto x _ {l}. \tag {100}
$$
These satisfy for all (parity) homogeneous elements  $x_{l}\in \mathrm{Cl}(V_{l},\mathfrak{q}_{l}),l = 1,2$  , the condition:
$$
\phi_ {1} \left(x _ {1}\right) \phi_ {2} \left(x _ {2}\right) = x _ {1} x _ {2} \stackrel {!} {=} (- 1) ^ {\operatorname {p r t} \left(x _ {1}\right) \cdot \operatorname {p r t} \left(x _ {2}\right)} \cdot x _ {2} x _ {1} = (- 1) ^ {\operatorname {p r t} \left(x _ {1}\right) \cdot \operatorname {p r t} \left(x _ {2}\right)} \cdot \phi_ {2} \left(x _ {2}\right) \phi_ {1} \left(x _ {1}\right). \tag {101}
$$
Indeed, we can by linearity reduce to the case that  $x_{1}$  and  $x_{2}$  are products of vectors  $v_{1} \in V_{1}$  and  $v_{2} \in V_{2}$ , resp. Since  $V_{1}$  and  $V_{2}$  are orthogonal to each other by assumption, for those elements we get by the identities D.3:
$$
v _ {1} v _ {2} = - v _ {2} v _ {1} + 2 \underbrace {\mathfrak {b} \left(v _ {1} , v _ {2}\right)} _ {= 0} = - v _ {2} v _ {1}. \tag {102}
$$
This shows the condition above for  $x_{1}$  and  $x_{2}$ .
We then define the  $\mathbb{F}$ -bilinear map:
$$
\operatorname {C l} \left(V _ {1}, \mathfrak {q} _ {1}\right) \times \operatorname {C l} \left(V _ {2}, \mathfrak {q} _ {2}\right)\rightarrow \operatorname {C l} (V, \mathfrak {q}), \quad \left(x _ {1}, x _ {2}\right) \mapsto x _ {1} x _ {2}, \tag {103}
$$
which thus factorizes through the  $\mathbb{F}$ -linear map:
$$
\phi : \operatorname {C l} \left(V _ {1}, \mathfrak {q} _ {1}\right) \hat {\otimes} \operatorname {C l} \left(V _ {2}, \mathfrak {q} _ {2}\right)\rightarrow \operatorname {C l} (V, \mathfrak {q}), \quad x _ {1} \hat {\otimes} x _ {2} \mapsto x _ {1} x _ {2}. \tag {104}
$$
We now show that  $\phi$  also respects multiplication. For this let  $x_{1},y_{1}\in \mathrm{Cl}(V_{1},\mathfrak{q}_{1})$  and  $x_{2},y_{2}\in \mathrm{Cl}(V_{2},\mathfrak{q}_{2})$  (parity) homogeneous elements. We then get:
$$
\begin{array}{l} \phi \left(\left(x _ {1} \hat {\otimes} x _ {2}\right) \cdot \left(y _ {1} \hat {\otimes} y _ {2}\right)\right) (105) \\ = \phi \left((- 1) ^ {\operatorname {p r t} \left(x _ {2}\right) \cdot \operatorname {p r t} \left(y _ {1}\right)} \cdot \left(x _ {1} y _ {1}\right) \hat {\otimes} \left(x _ {2} y _ {2}\right)\right) (106) \\ = (- 1) ^ {\operatorname {p r t} \left(x _ {2}\right) \cdot \operatorname {p r t} \left(y _ {1}\right)} \cdot x _ {1} y _ {1} x _ {2} y _ {2} (107) \\ = x _ {1} x _ {2} y _ {1} y _ {2} (108) \\ = \phi \left(x _ {1} \hat {\otimes} x _ {2}\right) \cdot \phi \left(y _ {1} \hat {\otimes} y _ {2}\right). (109) \\ \end{array}
$$
This shows the multiplicativity of  $\phi$  on homogeneous elements. The general case follows by the  $\mathbb{F}$ -linearity of  $\phi$ . Note that  $\phi$  also respects the parity grading.
Now consider the  $\mathbb{Z}/2\mathbb{Z}$ -graded  $\mathbb{F}$ -algebra  $\operatorname{Cl}(V_1, \mathfrak{q}_1) \hat{\otimes} \operatorname{Cl}(V_2, \mathfrak{q}_2)$  and  $\mathbb{Z}/2\mathbb{Z}$ -graded  $\mathbb{F}$ -algebra homomorphisms:
$$
\psi_ {1}: \operatorname {C l} \left(V _ {1}, \mathfrak {q} _ {1}\right)\rightarrow \operatorname {C l} \left(V _ {1}, \mathfrak {q} _ {1}\right) \hat {\otimes} \operatorname {C l} \left(V _ {2}, \mathfrak {q} _ {2}\right), \quad x _ {1} \mapsto x _ {1} \hat {\otimes} 1, \tag {110}
$$
$$
\psi_ {2}: \operatorname {C l} \left(V _ {2}, \mathfrak {q} _ {2}\right)\rightarrow \operatorname {C l} \left(V _ {1}, \mathfrak {q} _ {1}\right) \hat {\otimes} \operatorname {C l} \left(V _ {2}, \mathfrak {q} _ {2}\right), \quad x _ {2} \mapsto 1 \hat {\otimes} x _ {2}. \tag {111}
$$
Note that for all homogeneous elements  $x_{l}\in \mathrm{Cl}(V_{l},\mathfrak{q}_{l}),l = 1,2$  , we have:
$$
\psi_ {2} \left(x _ {2}\right) \cdot \psi_ {1} \left(x _ {1}\right) = \left(1 \hat {\otimes} x _ {2}\right) \cdot \left(x _ {1} \hat {\otimes} 1\right) \tag {112}
$$
$$
= (- 1) ^ {\operatorname {p r t} \left(x _ {1}\right) \cdot \operatorname {p r t} \left(x _ {2}\right)} \cdot x _ {1} \hat {\otimes} x _ {2} \tag {113}
$$
$$
= (- 1) ^ {\operatorname {p r t} \left(x _ {1}\right) \cdot \operatorname {p r t} \left(x _ {2}\right)} \cdot \left(x _ {1} \hat {\otimes} 1\right) \cdot \left(1 \hat {\otimes} x _ {2}\right) \tag {114}
$$
$$
= (- 1) ^ {\operatorname {p r t} \left(x _ {1}\right) \cdot \operatorname {p r t} \left(x _ {2}\right)} \cdot \psi_ {1} \left(x _ {1}\right) \cdot \psi_ {2} \left(x _ {2}\right). \tag {115}
$$
We then define the  $\mathbb{F}$ -linear map:
$$
\psi : V = V _ {1} \oplus V _ {2} \rightarrow \operatorname {C l} \left(V _ {1}, \mathfrak {q} _ {1}\right) \hat {\otimes} \operatorname {C l} \left(V _ {2}, \mathfrak {q} _ {2}\right), \quad v = v _ {1} + v _ {2} \mapsto \psi_ {1} \left(v _ {1}\right) + \psi_ {2} \left(v _ {2}\right) =: \psi (v). \tag {116}
$$
Note that we have:
$$
\psi (v) ^ {2} = \left(\psi_ {1} \left(v _ {1}\right) + \psi_ {2} \left(v _ {2}\right)\right) \cdot \left(\psi_ {1} \left(v _ {1}\right) + \psi_ {2} \left(v _ {2}\right)\right) \tag {117}
$$
$$
= \psi_ {1} \left(v _ {1}\right) ^ {2} + \psi_ {2} \left(v _ {2}\right) ^ {2} + \psi_ {1} \left(v _ {1}\right) \cdot \psi_ {2} \left(v _ {2}\right) + \psi_ {2} \left(v _ {2}\right) \cdot \psi_ {1} \left(v _ {1}\right) \tag {118}
$$
$$
= \psi_ {1} \left(v _ {1}\right) ^ {2} + \psi_ {2} \left(v _ {2}\right) ^ {2} + \underbrace {\psi_ {1} \left(v _ {1}\right) \cdot \psi_ {2} \left(v _ {2}\right) + (- 1) ^ {\operatorname {p r t} \left(v _ {1}\right) \cdot \operatorname {p r t} \left(v _ {2}\right)} \psi_ {1} \left(v _ {1}\right) \cdot \psi_ {2} \left(v _ {2}\right)} _ {= 0} \tag {119}
$$
$$
= \psi_ {1} \left(v _ {1} ^ {2}\right) + \psi_ {2} \left(v _ {2} ^ {2}\right) \tag {120}
$$
$$
= \mathfrak {q} _ {1} (v _ {1}) \cdot \psi_ {1} \left(1 _ {\operatorname {C l} \left(V _ {1}, \mathfrak {q} _ {1}\right)}\right) + \mathfrak {q} _ {2} (v _ {2}) \cdot \psi_ {2} \left(1 _ {\operatorname {C l} \left(V _ {2}, \mathfrak {q} _ {2}\right)}\right) \tag {121}
$$
$$
= \mathfrak {q} _ {1} (v _ {1}) \cdot 1 _ {\operatorname {C l} \left(V _ {1}, \mathfrak {q} _ {1}\right) \hat {\otimes} \operatorname {C l} \left(V _ {2}, \mathfrak {q} _ {2}\right)} + \mathfrak {q} _ {2} (v _ {2}) \cdot 1 _ {\operatorname {C l} \left(V _ {1}, \mathfrak {q} _ {1}\right) \hat {\otimes} \operatorname {C l} \left(V _ {2}, \mathfrak {q} _ {2}\right)} \tag {122}
$$
$$
= \left(\mathfrak {q} _ {1} \left(v _ {1}\right) + \mathfrak {q} _ {2} \left(v _ {2}\right)\right) \cdot 1 _ {\operatorname {C l} \left(V _ {1}, \mathfrak {q} _ {1}\right) \hat {\otimes}} \operatorname {C l} \left(V _ {2}, \mathfrak {q} _ {2}\right) \tag {123}
$$
$$
= \mathfrak {q} (v) \cdot 1 _ {\operatorname {C l} \left(V _ {1}, \mathfrak {q} _ {1}\right) \hat {\otimes}} \operatorname {C l} \left(V _ {2}, \mathfrak {q} _ {2}\right). \tag {124}
$$
By the universal property of the Clifford algebra  $\psi$  uniquely extends to an  $\mathbb{F}$ -algebra homomorphism: