SlowGuess/NeurIPS_2023_OCR huggingface dataset:
The advantages of more effective equivariant graph neural networks could be vast, particularly in the physical sciences. Improved equivariant neural networks could revolutionize fields like molecular biology, astrophysics, and materials science by accurately predicting system behaviors under various transformations. They could further enable more precise simulations of physical systems. Such new knowledge could drive forward scientific discovery, enabling innovations in healthcare, materials engineering, and our understanding of the universe. However, a significant drawback of relying heavily on advanced equivariant neural networks in physical sciences is the potential for unchecked errors or biases to propagate when these systems are not thoroughly cross-validated using various methods. This could potentially lead to flawed scientific theories or dangerous real-world applications.
Reproducibility Statement
We have released the code to reproduce the experimental results, including hyperparameters, network architectures, and so on, at https://github.com/DavidRuhe/clifford-group-equivariant-neural-networks.
Computational Resources
The volumetric quantities and regression experiments were carried out on  $1 \times 11$  GB NVIDIA GeForce GTX 1080 Ti and  $1 \times 11$  GB NVIDIA GeForce GTX 2080 Ti instances. The  $n$ -body experiment ran on  $1 \times 24$  GB NVIDIA GeForce RTX 3090 and  $1 \times 24$  GB NVIDIA RTX A5000 nodes. Finally, the top tagging experiment was conducted on  $4 \times 40$  GB NVIDIA Tesla A100 Ampere instances.
We would like to thank the scientific software development community, without whom this work would not be possible.
This work made use of Python (PSF License Agreement), NumPy [VDWCV11] (BSD-3 Clause), PyTorch [PGM+19] (BSD-3 Clause), CUDA (proprietary license), Weights and Biases [Bie20] (MIT), Scikit-Learn [PVG+11] (BSD-3 Clause), Seaborn [Was21] (BSD-3 Clause), matplotlib [Hun07] (PSF), [VGO+20] (BSD-3).
The top tagging dataset is licensed under CC-By Attribution 4.0 International.
1 Introduction 1
2 The Clifford Algebra 3
3 Theoretical Results 3
3.1 The Clifford Group and its Clifford Algebra Representations 4
3.2 Orthogonal Representations 6
4 Methodology 6
5 Experiments 8
5.1 Estimating Volumetric Quantities 8
5.2 O(5) Experiment: Regression 9
5.3 E(3) Experiment:  $n$ -Body System 9
5.4 O(1,3) Experiment: Top Tagging 10
6 Conclusion 10
References 11
Acknowledgements 17
Broader Impact Statement 17
Reproducibility Statement 17
Computational Resources 17
Licences 17
A Glossary 20
B Supplementary Material: Introduction 22
C Quadratic Vector Spaces and the Orthogonal Group 23
D The Clifford Algebra and Typical Constructions 26
D.1 The Clifford Algebra and its Universal Property 26
D.2 The Multivector Filtration and the Grade 27
D.3 The Parity Grading 28
D.4 The Dimension of the Clifford Algebra 29
D.5 Extending the Quadratic Form to the Clifford Algebra 32
D.6 The Multivector Grading 36
D.7 The Radical Subalgebra of the Clifford Algebra 44
D.8 The Center of the Clifford Algebra 45
D.9 The Twisted Center of the Clifford Algebra 48
E The Clifford Group and its Clifford Algebra Representations 49
E.1 Adjusting the Twisted Conjugation 50
E.2 The Clifford Group 54
E.3 The Structure of the Clifford Group 57
E.4 Orthogonal Representations of the Clifford Group 60
E.5 The Spinor Norm and the Clifford Norm 63
E.6 The Pin Group and the Spin Group 66
Notation	Meaning
O(n)	The orthogonal group acting on an n-dimensional vector space.
SO(n)	The special orthogonal group acting on an n-dimensional vector space.
E(n)	The Euclidean group acting on an n-dimensional vector space.
SE(n)	The special Euclidean group acting on an n-dimensional vector space.
GL(n)	The general linear group acting on an n-dimensional vector space.
O(V, q)	The orthogonal group of vector space V with respect to a quadratic form q.
O_R(V, q)	The orthogonal group of vector space V with respect to a quadratic form q that acts as the identity on R.
V	A vector space.
R	The radical vector subspace of V such that for any f ∈ R, b(f, v) = 0 for all v ∈ V.
F	A field.
R	The real numbers.
N	The natural numbers.
Z	The integers.
[n]	The set of integers {1, ..., n}.
q	A quadratic form, q: V → F.
b	A bilinear form, b: V × V → F.
Cl(V, q)	The Clifford algebra over a vector space V with quadratic form q.
Cl^×(V, q)	The group of invertible Clifford algebra elements.
Cl[x](V, q)	The group of invertible parity-homogeneous Clifford algebra elements. Cl[x](V, q) := {w ∈ Cl^×(V, q) | η(w) ∈ {±1}}.
Cl[0](V, q)	The even subalgebra of the Clifford algebra. Cl[0](V, q) := ∏m even n Cl(m)(V, q).
Cl[1](V, q)	The odd part of the Clifford algebra. Cl[1](V, q) := ∏m odd n Cl(m)(V, q).
Cl(m)(V, q)	The grade-m subspace of the Clifford algebra.
(-)^{m}	Grade projection, (−)^{m}: Cl(V, q) → Cl(m)(V, q).
ζ	Projection onto the zero grade, ζ: Cl(V, q) → Cl(0)(V, q).
ei	A basis vector, ei ∈ V.
fi	A basis vector of R.
eA	A Clifford basis element (product of basis vectors) with eA ∈ Cl(V, q), A ⊆ {1, ..., n}.
bar	The extended bilinear form on the Clifford algebra, bar: Cl(V, q) × Cl(V, q) → F.
bar	The extended quadratic form on the Clifford algebra, bar: Cl(V, q) → F.
α	Clifford main involution α: Cl(V, q) → Cl(V, q).
β	Main Clifford anti-involution, also known as reversion, β: Cl(V, q) → Cl(V, q).
γ	Clifford conjugation γ: Cl(V, q) → Cl(V, q).
η	Coboundary of α. For w ∈ Cl^×(V, q), η(w) ∈ {±1} if and only if w is parity-homogeneous.
ρ(w)	The (adjusted) twisted conjugation, used as the action of the Clifford group. ρ(w): Cl(V, q) → Cl(V, q), w ∈ Cl^×(V, q).
Γ(V, q)	The Clifford group of Cl(V, q).
Γ[0](V, q)	The special Clifford group. It excludes orientation-reversing (odd) elements. Γ[0](V, q) := Γ(V, q) ∩ Cl[0](V, q).
∧(R)	The radical subalgebra of Cl(V, q). I.e., those elements that have zero q. It is equal to the exterior algebra of R.
∧[i](R)	The even or odd (i ∈ {0,1}) subalgebra of ∧(R). ∧[i](R) := ∧(R) ∩ Cl[i](V, q).
∧(≥1)(R)	The subalgebra of ∧(R) with grade greater than or equal to one. ∧(≥1)(R) := span{f1...fk | k ≥ 1, fl ∈ R}.
∧×(R)	The group of invertible elements of ∧(R). ∧×(R) = F× + ∧(≥1)(R).
∧[×](R)	The group of invertible elements of ∧(R) with even grades. ∧[×](R) := F× + span{f1...fk | k ≥ 2 even, fl ∈ R}.
∧* (R)	Same as ∧×(R), but with F× set to 1.
∧[∗](R)	Same as ∧[×](R), but with F× set to 1.
SN	Spinor norm, SN: Cl(V, q) → F.
CN	Clifford norm, CN: Cl(V, q) → F.
B Supplementary Material: Introduction
Existing literature on Clifford algebra uses varying notations, conventions, and focuses on several different applications. We gathered previous definitions and included independently derived results to achieve the desired outcomes for this work and to provide proofs for the most general cases, including, e.g., potential degeneracy of the metric. As such, this appendix acts as a comprehensive resource on quadratic spaces, orthogonal groups, Clifford algebras, their constructions, and specific groups represented within Clifford algebras.
We start in Appendix C with a primer to quadratic spaces and the orthogonal group. In particular, we specify how the orthogonal group is defined on a quadratic space, and investigate its action. We pay special attention to the case of spaces with degenerate quadratic forms, and what we call "radical-preserving" orthogonal automorphisms. The section concludes with the presentation of the Cartan-Dieudonné theorem.
In Appendix D, we introduce the definition of the Clifford algebra as a quotient of the tensor algebra. We investigate several key properties of the Clifford algebra. Then, we introduce its parity grading, which in turn is used to prove that the dimension of the algebra is  $2^n$ , where  $n$  is the dimension of the vector space on which it is defined. This allows us to construct an algebra basis. Subsequently, we extend the quadratic and bilinear forms of the vector space to their Clifford algebra counterparts. This then leads to the construction of an orthogonal basis for the algebra. Furthermore, the multivector grading of the algebra is shown to be basis independent, leading to the orthogonal sum decomposition into the usual subspaces commonly referred to as scalars, vectors, bivectors, and so on. Finally, we investigate some additional properties of the algebra, such as its center, its radical subalgebra, and its twisted center.
In Appendix E we introduce, motivate, and adjust the twisted conjugation map. We show that it comprises an algebra automorphism, thereby respecting the algebra's vector space and also its multiplicative properties. Moreover, it can serve as a representation of the Clifford algebra's group of invertible elements. We then identify what we call the Clifford group, whose action under the twisted conjugation respects also the multivector decomposition. We investigate properties of the Clifford group and its action. Specifically, we show that its action yields a radical-preserving orthogonal automorphism. Moreover, it acts orthogonally on each individual subspace.
Finally, after introducing the Spinor and Clifford norm, the section concludes with the pursuit of a general definition for the Pin and Spin group, which are used in practice more often than the Clifford group. This, however, turns out to be somewhat problematic when generalizing to fields beyond  $\mathbb{R}$ . We motivate several choices, and outline their (dis)advantages. For  $\mathbb{R}$ , we finally settle on definitions that are compatible with existing literature.
C Quadratic Vector Spaces and the Orthogonal Group
We provide a short introduction to quadratic spaces and define the orthogonal group of a (non-definite) quadratic space. We use these definitions in our analysis of how the Clifford group relates to the orthogonal group.
We will always denote with  $\mathbb{F}$  a field of a characteristic different from 2,  $\mathrm{char}(\mathbb{F})\neq 2$ . Sometimes we will specialize to the real numbers  $\mathbb{F} = \mathbb{R}$ . Let  $V$  be a vector space over  $\mathbb{F}$  of finite dimension  $\dim_{\mathbb{F}}V = n$ . We will follow [Ser12].
Definition C.1 (Quadratic forms and quadratic vector spaces). A map  $\mathfrak{q}: V \to \mathbb{F}$  will be called a quadratic form of  $V$  if for all  $c \in \mathbb{F}$  and  $v \in V$ :
$$
\mathfrak {q} (c \cdot v) = c ^ {2} \cdot \mathfrak {q} (v), \tag {17}
$$
and if:
$$
\mathfrak {b} \left(v _ {1}, v _ {2}\right) := \frac {1}{2} \left(\mathfrak {q} \left(v _ {1} + v _ {2}\right) - \mathfrak {q} \left(v _ {1}\right) - \mathfrak {q} \left(v _ {2}\right)\right), \tag {18}
$$
is a bilinear form over  $\mathbb{F}$  in  $v_{1}, v_{2} \in V$ , i.e., it is separately  $\mathbb{F}$ -linear in each of the arguments  $v_{1}$  and  $v_{2}$  when the other one is fixed.
The tuple  $(V,\mathfrak{q})$  will then be called a quadratic (vector) space.
Remark C.2. 1. Note that we explicitly do not make assumptions about the non-degeneracy of  $\mathfrak{b}$ . Even the extreme case with constant  $\mathfrak{q} = 0$  is allowed and of interest.
2. Further note, that  $\mathfrak{b}$  will automatically be a symmetric bilinear form.
Definition C.3 (The radical subspace). Now consider the quadratic space  $(V, \mathfrak{q})$ . We then call the subspace:
$$
\mathcal {R} := \{f \in V | \forall v \in V. \mathfrak {b} (f, v) = 0 \}, \tag {19}
$$
the radical subspace of  $(V,\mathfrak{q})$
Remark C.4. 1. The radical subspace of  $(V, \mathfrak{q})$  is the biggest subspace of  $V$  where  $\mathfrak{q}$  is degenerate. Note that this space is orthogonal to all other subspaces of  $V$ .
2. If  $W$  is any complementary subspace of  $\mathcal{R}$  in  $V$ , so that  $V = \mathcal{R} \oplus^{\perp} W$ , then  $\mathfrak{q}$  restricted to  $W$  is non-degenerate.
Definition C.5 (Orthogonal basis). A basis  $e_1, \ldots, e_n$  of  $V$  is called orthogonal basis of  $V$  if for all  $i \neq j$  we have:
$$
\mathfrak {b} \left(e _ {i}, e _ {j}\right) = 0. \tag {20}
$$
It is called an orthonormal basis if, in addition,  $\mathfrak{q}(e_i) \in \{-1, 0, +1\}$  for all  $i = 1, \dots, n$ .
Remark C.6. Note that every quadratic space  $(V,\mathfrak{q})$  has an orthogonal basis by [Ser12] p. 30 Thm. 1, but not necessarily a orthonormal basis. However, if  $\mathbb{F} = \mathbb{R}$  then  $(V,\mathfrak{q})$  has an orthonormal basis by Sylvester's law of inertia, see [Syl52].
Definition C.7 (The orthogonal group). For a quadratic space  $(V, \mathfrak{q})$  we define the orthogonal group of  $(V, \mathfrak{q})$  as follows:
$$
\mathrm {O} (\mathfrak {q}) := \mathrm {O} (V, \mathfrak {q}) := \left\{\Phi : V \rightarrow V \mid \Phi \text {F - l i n e a r a t o m o r p h i s m} ^ {8}, \text {s . t .} \forall v \in V. \mathfrak {q} (\Phi (v)) = \mathfrak {q} (v) \right\}. \tag {21}
$$
If  $(V,\mathfrak{q}) = \mathbb{R}^{(p,q,r)}$ , i.e. if  $\mathbb{F} = \mathbb{R}$  and  $V = \mathbb{R}^n$  and  $\mathfrak{q}$  has the signature  $(p,q,r)$  with  $p + q + r = n$  then we define the group of orthogonal matrices of signature  $(p,q,r)$  as follows:
$$
O (p, q, r) := \left\{O \in \operatorname {G L} (n) \mid O ^ {\top} \Delta_ {(p, q, r)} O = \Delta_ {(p, q, r)} \right\}, \tag {22}
$$
where we used the  $(n\times n)$  -diagonal signature matrix:
$$
\Delta_ {(p, q, r)} := \operatorname {d i a g} \left(\underbrace {+ 1 , \dots , + 1} _ {p - t i m e s}, \underbrace {- 1 , \dots , - 1} _ {q - t i m e s}, \underbrace {0 , \dots , 0} _ {r - t i m e s}\right). \tag {23}
$$
Theorem C.8 (See [YCo20]). Let  $(V, \mathfrak{q})$  be a finite dimensional quadratic space over a field  $\mathbb{F}$  with  $\operatorname{char}(\mathbb{F}) \neq 2$ . Let  $\mathcal{R} \subseteq V$  be the radical subspace,  $r \coloneqq \dim \mathcal{R}$ , and,  $W \subseteq V$  a complementary subspace,  $m \coloneqq \dim W$ . Then we get an isomorphism:
$$
\mathrm {O} (V, \mathfrak {q}) \cong \left( \begin{array}{c c} \mathrm {O} (W, \mathfrak {q} | _ {W}) & 0 _ {m \times r} \\ \mathrm {M} (r, m) & \mathrm {G L} (r) \end{array} \right) = \left\{\left( \begin{array}{c c} O & 0 _ {m \times r} \\ M & G \end{array} \right) \mid O \in \mathrm {O} (W, \mathfrak {q} | _ {W}), M \in \mathrm {M} (r, m), G \in \mathrm {G L} (r) \right\}, \tag {24}
$$
where  $\mathrm{M}(r,m) := \mathbb{F}^{r\times m}$  is the additive group of all  $(r\times m)$ -matrices with coefficients in  $\mathbb{F}$  and where  $\operatorname{GL}(r)$  is the multiplicative group of all invertible  $(r\times r)$ -matrices with coefficients in  $\mathbb{F}$ .
Proof. Let  $e_1, \ldots, e_m$  be an orthogonal basis for  $W$  and  $f_1, \ldots, f_r$  be a basis for  $\mathcal{R}$ , then the associated bilinear form  $\mathfrak{b}$  of  $\mathfrak{q}$  has the following matrix representation:
$$
\left( \begin{array}{c c} Q & 0 \\ 0 & 0 \end{array} \right), \tag {25}
$$
with an invertible diagonal  $(m\times m)$ -matrix  $Q$ . For the matrix of any orthogonal automorphism  $\Phi$  of  $V$  we get the necessary and sufficient condition:
$$
\begin{array}{l} \left( \begin{array}{c c} Q & 0 \\ 0 & 0 \end{array} \right) \stackrel {!} {=} \left( \begin{array}{c c} A ^ {\top} & C ^ {\top} \\ B ^ {\top} & D ^ {\top} \end{array} \right) \left( \begin{array}{c c} Q & 0 \\ 0 & 0 \end{array} \right) \left( \begin{array}{c c} A & B \\ C & D \end{array} \right) (26) \\ = \left( \begin{array}{l l} A ^ {\top} Q A & A ^ {\top} Q B \\ B ^ {\top} Q A & B ^ {\top} Q B \end{array} \right). (27) \\ \end{array}
$$
This is equivalent to the conditions:
$$
Q = A ^ {\top} Q A, \quad 0 = A ^ {\top} Q B, \quad 0 = B ^ {\top} Q B. \tag {28}
$$
This shows that  $A$  is the matrix of an orthogonal automorphism of  $W$  (w.r.t.  $\mathfrak{b}|_W$ ), which, since  $Q$  is invertible, is also invertible. The second equation then shows that necessarily  $B = 0$ . Since the whole matrix needs to be invertible also  $D$  must be invertible. Furthermore, there are no constraints on  $C$ .
If all those conditions are satisfied the whole matrix satisfies the orthogonality constraints from above.
Corollary C.9. For  $\mathbb{R}^{(p,q,r)}$  we get:
$$
O (p, q, r) \cong \left( \begin{array}{c c} O (p, q) & 0 _ {(p + q) \times r} \\ M (r, p + q) & G L (r) \end{array} \right), \tag {29}
$$
where  $\mathrm{O}(p,q)\coloneqq \mathrm{O}(p,q,0)$
Remark C.10. Note that the composition  $\Phi_1\circ \Phi_2$  of orthogonal automorphisms of  $(V,\mathfrak{q})$  corresponds to the matrix multiplication as follows:
$$
\left( \begin{array}{c c} O _ {1} & 0 \\ M _ {1} & G _ {1} \end{array} \right) \left( \begin{array}{c c} O _ {2} & 0 \\ M _ {2} & G _ {2} \end{array} \right) = \left( \begin{array}{c c} O _ {1} O _ {2} & 0 \\ M _ {1} O _ {2} + G _ {1} M _ {2} & G _ {1} G _ {2} \end{array} \right). \tag {30}
$$
Definition C.11 (Radical preserving orthogonal automorphisms). For a quadratic space  $(V, \mathfrak{q})$  with radical subspace  $\mathcal{R} \subseteq V$  we define the group of radical preserving orthogonal automorphisms as follows:
$$
\mathrm {O} _ {\mathcal {R}} (V, \mathfrak {q}) := \left\{\Phi \in \mathrm {O} (V, \mathfrak {q}) \mid \Phi | _ {\mathcal {R}} = \mathrm {i d} _ {\mathcal {R}} \right\}. \tag {31}
$$
If  $(V,\mathfrak{q}) = \mathbb{R}^{(p,q,r)}$ , i.e. if  $\mathbb{F} = \mathbb{R}$  and  $V = \mathbb{R}^n$  and  $\mathfrak{q}$  has the signature  $(p,q,r)$  with  $p + q + r = n$  then we define the group of radical preserving orthogonal matrices of signature  $(p,q,r)$  as follows:
$$
\mathrm {O} _ {\mathcal {R}} (p, q, r) := \left\{O \in \mathrm {O} (p, q, r) \mid \text {b o t t o m r i g h t c o r n e r o f} O = I _ {r} \right\} = \left( \begin{array}{c c} \mathrm {O} (p, q) & 0 _ {(p + q) \times r} \\ \mathrm {M} (r, p + q) & I _ {r} \end{array} \right), \tag {32}
$$
where  $I_r$  is the  $(r \times r)$ -identity matrix.
Remark C.12. Note that the matrix representation of  $\Phi \in \mathrm{O}_{\mathcal{R}}(\mathfrak{q})$  w.r.t. an orthogonal basis like in the proof of theorem C.8 is of the form:
$$
\left( \begin{array}{c c} O & 0 _ {m \times r} \\ M & I _ {r} \end{array} \right), \tag {33}
$$
with  $O \in \mathrm{O}(W, \mathfrak{q}|_W)$ ,  $M \in \mathrm{M}(r, m)$  and where  $I_r$  is the  $(r \times r)$ -identity matrix.
The composition  $\Phi_1\circ \Phi_2$  of  $\Phi_{1}$  and  $\Phi_2\in \mathrm{O}_{\mathcal{R}}(\mathfrak{q})$  is then given by the corresponding matrix multiplication:
$$
\left( \begin{array}{l l} O _ {1} & 0 \\ M _ {1} & I _ {r} \end{array} \right) \left( \begin{array}{l l} O _ {2} & 0 \\ M _ {2} & I _ {r} \end{array} \right) = \left( \begin{array}{l l} O _ {1} O _ {2} & 0 \\ M _ {1} O _ {2} + M _ {2} & I _ {r} \end{array} \right). \tag {34}
$$
By observing the left column (the only part that does not transform trivially), we see that  $\mathrm{O}(\mathfrak{q}|_W)$  acts on  $\mathrm{M}(r,m)$  just by matrix multiplication from the right (in the corresponding basis):
$$
\left(O _ {1}, M _ {1}\right) \cdot \left(O _ {2}, M _ {2}\right) = \left(O _ {1} O _ {2}, M _ {1} O _ {2} + M _ {2}\right). \tag {35}
$$
This immediately shows that we can write  $\mathrm{O}_{\mathcal{R}}(\mathfrak{q})$  as the semi-direct product:
$$
\mathrm {O} _ {\mathcal {R}} (\mathfrak {q}) \cong \mathrm {O} (\mathfrak {q} | _ {W}) \ltimes \mathrm {M} (r, m). \tag {36}
$$
In the special case of  $\mathbb{R}^{(p,q,r)}$  we get:
$$
\mathrm {O} _ {\mathcal {R}} (p, q, r) \cong \mathrm {O} (p, q) \ltimes \mathrm {M} (r, p + q). \tag {37}
$$
We conclude this chapter by citing the Theorem of Cartan and Dieudonné about the structure of orthogonal groups in the non-degenerate case, but still for arbitrary fields  $\mathbb{F}$  with  $\mathrm{char}(\mathbb{F})\neq 2$
Theorem C.13 (Theorem of Cartan-Dieudonné, see [Art57] Thm. 3.20). Let  $(V,\mathfrak{q})$  be a nondegenerate quadratic space of finite dimension  $\dim V = n < \infty$  over a field  $\mathbb{F}$  of  $\mathrm{char}(\mathbb{F})\neq 2$ . Then every element  $g\in \mathrm{O}(V,\mathfrak{q})$  can be written as:
$$
g = r _ {1} \circ \dots \circ r _ {k}, \tag {38}
$$
with  $1 \leq k \leq n$ , where  $r_i$  are reflections w.r.t. non-singular hyperplanes.
D The Clifford Algebra and Typical Constructions
In this section, we provide the required definitions, constructions, and derivations leading to the results stated in the main paper. We start with a general introduction to the Clifford algebra.
D.1 The Clifford Algebra and its Universal Property
We follow [LS09, Cru90].
Let  $(V,\mathfrak{q})$  be a finite dimensional quadratic space over a field  $\mathbb{F}$  (also denoted an  $\mathbb{F}$ -vector space) with  $\operatorname{char}(\mathbb{F})\neq 2$ . We abbreviate the corresponding bilinear form  $\mathfrak{b}$  on vectors  $v_{1},v_{2}\in V$  as follows:
$$
\mathfrak {b} \left(v _ {1}, v _ {2}\right) := \frac {1}{2} \left(\mathfrak {q} \left(v _ {1} + v _ {2}\right) - \mathfrak {q} \left(v _ {1}\right) - \mathfrak {q} \left(v _ {2}\right)\right). \tag {39}
$$
Definition D.1. To define the Clifford algebra  $\operatorname{Cl}(V, \mathfrak{q})$  we first consider the tensor algebra of  $V$ :
$$
\mathrm {T} (V) := \bigoplus_ {m = 0} ^ {\infty} V ^ {\otimes m} = \operatorname {s p a n} \left\{v _ {1} \otimes \dots \otimes v _ {m} \mid m \geq 0, v _ {i} \in V \right\}, \tag {40}
$$
$$
V ^ {\otimes m} := \underbrace {V \otimes \cdots \otimes V} _ {m - t i m e s}, \quad V ^ {\otimes 0} := \mathbb {F}, \tag {41}
$$
and the following two-sided ideal:
$$
I (\mathfrak {q}) := \left\langle v \otimes v - \mathfrak {q} (v) \cdot 1 _ {\mathrm {T} (V)} \mid v \in V \right\rangle . \tag {42}
$$
Then we define the Clifford algebra  $\operatorname{Cl}(V, \mathfrak{q})$  as the following quotient:
$$
\operatorname {C l} (V, \mathfrak {q}) := \mathrm {T} (V) / I (\mathfrak {q}). \tag {43}
$$
In words, we identify the square of a vector with its quadratic form. We also denote the canonical algebra quotient map as:
$$
\pi : \mathrm {T} (V) \rightarrow \operatorname {C l} (V, \mathfrak {q}). \tag {44}
$$
Remark D.2. 1. It is not easy to see, but always true, that  $\dim \operatorname{Cl}(V, \mathfrak{q}) = 2^n$ , where  $n := \dim V$ , see Theorem D.15.
2. If  $e_1, \ldots, e_n$  is any basis of  $(V, \mathfrak{q})$  then  $(e_A)_{A \subseteq [n]}$  is a basis for  $\operatorname{Cl}(V, \mathfrak{q})$ , where we put for a subset  $A \subseteq [n] := \{1, \ldots, n\}$ :
$$
e _ {A} := \prod_ {i \in A} ^ {<  } e _ {i}, \quad e _ {\emptyset} := 1 _ {\operatorname {C l} (V, \mathfrak {q})}. \tag {45}
$$
where the product is taken in increasing order of the indices  $i \in A$ , see Corollary D.16.
3. If  $e_1, \ldots, e_n$  is any orthogonal basis of  $(V, \mathfrak{q})$ , then one can even show that  $(e_A)_{A \subseteq [n]}$  is an orthogonal basis for  $\operatorname{Cl}(V, \mathfrak{q})$  w.r.t. an extension of the bilinear form  $\mathfrak{b}$  from  $V$  to  $\operatorname{Cl}(V, \mathfrak{q})$ , see Theorem D.26.
Lemma D.3 (The fundamental identity). Note that for  $v_{1}, v_{2} \in V$ , we always have the fundamental identity in  $\operatorname{Cl}(V, \mathfrak{q})$ :
$$
v _ {1} v _ {2} + v _ {2} v _ {1} = 2 \mathfrak {b} \left(v _ {1}, v _ {2}\right). \tag {46}
$$
Proof. By definition of the Clifford algebra we have the identities:
$$
\begin{array}{l} \mathfrak {q} \left(v _ {1}\right) + v _ {1} v _ {2} + v _ {2} v _ {1} + \mathfrak {q} \left(v _ {2}\right) = v _ {1} v _ {1} + v _ {1} v _ {2} + v _ {2} v _ {1} + v _ {2} v _ {2} (47) \\ = \left(v _ {1} + v _ {2}\right) \left(v _ {1} + v _ {2}\right) (48) \\ = \mathfrak {q} \left(v _ {1} + v _ {2}\right) (49) \\ = \mathfrak {b} \left(v _ {1} + v _ {2}, v _ {1} + v _ {2}\right) (50) \\ = \mathfrak {b} \left(v _ {1}, v _ {1}\right) + \mathfrak {b} \left(v _ {1}, v _ {2}\right) + \mathfrak {b} \left(v _ {2}, v _ {1}\right) + \mathfrak {b} \left(v _ {2}, v _ {2}\right) (51) \\ = \mathfrak {q} (v _ {1}) + 2 \mathfrak {b} (v _ {1}, v _ {2}) + \mathfrak {q} (v _ {2}). (52) \\ \end{array}
$$
Subtracting  $\mathfrak{q}(v_1) + \mathfrak{q}(v_2)$  on both sides gives the claim.
The ideal ensures that for all elements containing  $v \otimes v$  (i.e., linear combinations, multiplications on the left, and multiplications on the right),  $v \otimes v$  is identified with  $\mathfrak{q}(v)$ ; e.g.  $x \otimes v \otimes v \otimes y \sim x \otimes (\mathfrak{q}(v) \cdot 1_{\mathrm{T}(V)}) \otimes y$  for every  $x, y \in \mathrm{T}(V)$ .