The Low-Dimensional Classification

Introduction

This article classifies the Clifford algebras of a non-degenerate quadratic form in dimensions one, two and three over a field, and records the two operations — complexification and realification — that pass between the real and the complex classification. The general theory of the Clifford algebra is taken as given from the companion layer: $\mathrm{Cl}(V,q)$ is the quotient of the tensor algebra $T(V)$ by the two-sided ideal generated by the elements $v\otimes v-q(v)\cdot 1$, it is $\mathbb{Z}/2$-graded as $\mathrm{Cl}(V,q)=\mathrm{Cl}^0(V,q)\oplus\mathrm{Cl}^1(V,q)$, its multiplication obeys the fundamental relation

$$ uv+vu=2g(u,v)\cdot 1, $$

and the algebra of an orthogonal direct sum is the graded tensor product

$$ \mathrm{Cl}(V_1\oplus V_2,\,q_1\perp q_2)\cong \mathrm{Cl}(V_1,q_1)\,\hat{\otimes}\,\mathrm{Cl}(V_2,q_2). $$

That material is established in the opening of the Clifford layer and is used here without restatement. The present article adds only the computation of the small cases and the two base-change operations.

The base structure is a field $F$ of characteristic not $2$, so that $2$ is invertible and the polar form $g(u,v)=\tfrac12\bigl(q(u+v)-q(u)-q(v)\bigr)$ is available. Most statements hold over any commutative ring in which $2$ is a unit; where a field is needed for a diagonalisation or for an isomorphism of simple algebras, this is said explicitly. The results over a general commutative ring are not covered here of the category; the reader should read the field case here as the model that the ring case generalises.

The real signature convention is fixed once. The algebra $\mathrm{Cl}_{p,q}$ is generated by an orthogonal family $e_1,\dots,e_{p+q}$ with

$$ e_i^2=+1 \quad (1\le i\le p), \qquad e_i^2=-1 \quad (p

Thus $p$ counts the generators of positive square. This is the convention under which $\mathrm{Cl}_{0,1}\cong\mathbb{C}$ and $\mathrm{Cl}_{1,0}\cong\mathbb{D}$, and it is the one held throughout the category.

One Dimension

The one-dimensional cases are the atoms from which every other case is assembled, because a diagonalised form is an orthogonal sum of one-dimensional forms and the graded tensor product then multiplies the algebras.

The negative generator. Let $V=Fe$ with $q(e)=-1$. The defining relation is $e^2=-1$, so the algebra is the quotient of the polynomial algebra by $x^2+1$:

$$ \mathrm{Cl}_{0,1}=F[e]/(e^2+1). $$

If $-1$ is not a square in $F$, this is the quadratic field extension $F(\sqrt{-1})$, and in particular $\mathrm{Cl}_{0,1}\cong\mathbb{C}$ when $F=\mathbb{R}$. The algebra is commutative, two-dimensional, and a field when $-1$ is not a square.

The positive generator. Let $V=Fe$ with $q(e)=+1$. The defining relation is $e^2=+1$, so

$$ \mathrm{Cl}_{1,0}=F[e]/(e^2-1). $$

When $2$ is invertible the polynomial $x^2-1=(x-1)(x+1)$ has the two coprime factors $x-1$ and $x+1$, so by the Chinese remainder theorem

$$ \mathrm{Cl}_{1,0}\cong F\times F, $$

with the two idempotents

$$ \pi_+=\tfrac12(1+e), \qquad \pi_-=\tfrac12(1-e), \qquad \pi_+^2=\pi_+,\quad \pi_-^2=\pi_-, \quad \pi_+\pi_-=0. $$

The algebra generated by $\pi_+$ and $\pi_-$ is isomorphic to the split complex algebra $\mathbb{D}=F\times F$, and for $F=\mathbb{R}$ this is $\mathbb{R}\times\mathbb{R}=\mathbb{D}$. So $\mathrm{Cl}_{1,0}\cong\mathbb{D}$. The pair $\mathrm{Cl}_{0,1}\cong\mathbb{C}$ and $\mathrm{Cl}_{1,0}\cong\mathbb{D}$ is the origin of the entire dependence of the classification on the signs: a single sign flip in the form interchanges the field with the split algebra.

Remark. The two algebras are exchanged by replacing $q$ with $-q$, that is, by replacing $e$ with an element of square $-1$ in place of one of square $+1$. Since the complex and the split complex algebra are not isomorphic over $\mathbb{R}$, the classification is genuinely sensitive to the signs of the form and not merely to its dimension. This is the finite-dimensional shadow of the fact that the order of the form, and not only its rank, is an invariant.

Two Dimensions

The two-dimensional cases are obtained from the one-dimensional ones by the graded tensor product, or directly.

The definite negative case. Let $e_1,e_2$ satisfy $e_1^2=e_2^2=-1$ and $e_1e_2=-e_2e_1$, and write $e_3=e_1e_2$. Then $e_3^2=(e_1e_2)(e_1e_2)=-e_1^2e_2^2=-(-1)(-1)=-1$, while

$$ e_1e_2=e_3,\qquad e_2e_1=-e_3,\qquad e_1e_3=e_1(e_1e_2)=-e_2,\qquad e_3e_1=(e_1e_2)e_1=e_2. $$

So $e_1,e_2,e_3$ obey the quaternion relations of the category, $e_k^2=-1$ for each $k$ together with the cyclic products $e_1e_2=e_3$, $e_2e_3=e_1$, $e_3e_1=e_2$, and

$$ \mathrm{Cl}_{0,2}\cong\mathbb{H}. $$

This is the first non-commutative case, and the presentation is that of the quaternion algebra $(-1,-1)_F$: its center is $F$, its dimension is $4$, and it is a division algebra exactly when the form $\operatorname{diag}(-1,-1)$ is anisotropic, which holds over $\mathbb{R}$. Over a field whose quaternion algebra splits — for instance a field containing a square root of $-1$ — the same relations give $M_2(F)$ instead, and the division-algebra statement is then false; the sharpening of the criterion over a general field belongs to the Witt-theoretic layer.

The split case. Let $e_1^2=+1$, $e_2^2=-1$, with $e_1e_2=-e_2e_1$. The assignment

$$ e_1\longmapsto \begin{pmatrix}1&0\\0&-1\end{pmatrix}, \qquad e_2\longmapsto \begin{pmatrix}0&-1\\1&0\end{pmatrix} $$

has image a pair of matrices that square to $+1$ and $-1$ respectively and anticommute:

$$ \begin{pmatrix}1&0\\0&-1\end{pmatrix}^2 =\begin{pmatrix}1&0\\0&1\end{pmatrix},\qquad \begin{pmatrix}0&-1\\1&0\end{pmatrix}^2 =\begin{pmatrix}-1&0\\0&-1\end{pmatrix},\qquad \begin{pmatrix}1&0\\0&-1\end{pmatrix}\begin{pmatrix}0&-1\\1&0\end{pmatrix} =-\begin{pmatrix}0&-1\\1&0\end{pmatrix}\begin{pmatrix}1&0\\0&-1\end{pmatrix}. $$

By the universal property the assignment extends to an algebra homomorphism $\mathrm{Cl}_{1,1}\to M_2(F)$; it is injective because the images generate $M_2(F)$ and both algebras have dimension $4$. Hence

$$ \mathrm{Cl}_{1,1}\cong M_2(F). $$

This four-dimensional algebra is the one called the split quaternion algebra in the classical literature, and over $\mathbb{R}$ it is the algebra of in a different presentation: that article defines its algebra as the tensor product $\mathbb{H}_{\mathbb{D}}=\mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}$, of real dimension eight, so that $\mathbb{H}_{\mathbb{D}}\cong\mathbb{H}\times\mathbb{H}$ and not $M_2(\mathbb{R})$. The two algebras are distinct and carry the same name in the literature; in this corpus the symbol $\mathbb{H}_{\mathbb{D}}$ denotes the eight-dimensional tensor product throughout, and $\mathrm{Cl}_{1,1}$ is written $M_2(F)$.

The definite positive case. Let $e_1^2=e_2^2=+1$, $e_1e_2=-e_2e_1$. The assignment

$$ e_1\longmapsto \begin{pmatrix}1&0\\0&-1\end{pmatrix}, \qquad e_2\longmapsto \begin{pmatrix}0&1\\1&0\end{pmatrix} $$

again has the two squares equal to $+1$ and the two factors anticommuting, so by the same argument

$$ \mathrm{Cl}_{2,0}\cong M_2(F). $$

So $\mathrm{Cl}_{1,1}$ and $\mathrm{Cl}_{2,0}$ are isomorphic, while $\mathrm{Cl}_{0,2}$ is the quaternion algebra. The first two are split (they contain zero divisors and are full matrix algebras), and the third is a division algebra. The sign change $q\mapsto -q$ replaces each generator by one of the opposite square; it preserves the dimension but not the isomorphism class, and the two algebras it relates, $M_2(F)$ and $\mathbb{H}$, are the two real forms of the same complex algebra $\mathbb{C}\mathrm{l}_2$. Their classes in the eightfold table differ by $d\mapsto-d$, and their complexifications agree.

Remark. All three algebras are four-dimensional. The classification in dimension two is therefore not a classification by dimension but by the type of the algebra — division, split $2\times2$ matrices, or quaternion — and it is the determinant of the form that decides between them.

Three Dimensions

The case $\mathrm{Cl}_{0,3}$. Let $e_1,e_2,e_3$ all square to $-1$ and anticommute. The volume element $\omega=e_1e_2e_3$ satisfies

$$ \omega^2=(-1)^{3\cdot 2/2}(-1)^3=(-1)^3\cdot(-1)=+1, $$

using the formula $\omega^2=(-1)^{n(n-1)/2}q(e_1)\cdots q(e_n)$ for an orthogonal basis. Hence

$$ f_\pm=\tfrac12(1\pm\omega), \qquad f_+^2=f_+,\quad f_-^2=f_-,\quad f_+f_-=0,\quad f_++f_-=1 $$

are two orthogonal central idempotents, and $\omega$ commutes with every generator because $n=3$ is odd. The algebra therefore splits as

$$ \mathrm{Cl}_{0,3}\cong \mathrm{Cl}_{0,3}f_+\times \mathrm{Cl}_{0,3}f_-. $$

Each factor is generated by the three elements $e_1f_\pm,e_2f_\pm,e_3f_\pm$, which still square to $-1$ and anticommute, so each factor is a copy of $\mathbb{H}$. Hence

$$ \mathrm{Cl}_{0,3}\cong\mathbb{H}\times\mathbb{H}. $$

The algebra is semisimple but not simple, and its center is two-dimensional.

The case $\mathrm{Cl}_{3,0}$. Now the three generators square to $+1$. The Pauli matrices

$$ \sigma_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad \sigma_2=\begin{pmatrix}0&-i\\ i&0\end{pmatrix},\qquad \sigma_3=\begin{pmatrix}1&0\\0&-1\end{pmatrix} $$

satisfy $\sigma_k^2=I$ and $\sigma_j\sigma_k=-\sigma_k\sigma_j$ for $j\neq k$. By the universal property the assignment $e_k\mapsto\sigma_k$ extends to a homomorphism of $F$-algebras $\mathrm{Cl}_{3,0}\to M_2(F(\sqrt{-1}))$, over a field $F$ of characteristic not $2$ in which $-1$ is not a square, so that $F(\sqrt{-1})$ is two-dimensional over $F$; it is surjective, because the images together with the identity span the four-dimensional algebra $M_2(F(\sqrt{-1}))$ over $F(\sqrt{-1})$, and both sides have dimension $8$ over $F$. Therefore

$$ \mathrm{Cl}_{3,0}\cong M_2(F(\sqrt{-1}))=\mathbb{C}\mathrm{l}_2, $$

the last identification being the standard isomorphism of the complex Clifford algebra $\mathbb{C}\mathrm{l}_2$ with the complex $2\times2$ matrices, understood over $F=\mathbb{R}$, where the hypothesis on $-1$ holds. When $-1$ is a square in $F$ the volume element can be rescaled to square $+1$, the center becomes $F\times F$, and the algebra splits; the split entries of the table below then apply.

This is the algebra of: $M_2(\mathbb{C})$ is also $\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}=\mathbb{B}$. The identification is made explicit by the quaternion units. Writing $\gamma_k$ for the generators of $\mathrm{Cl}_{3,0}$ and $e_k$ for the quaternion units of $\mathbb{B}$, the map

$$ \gamma_k\longmapsto ie_k \qquad (k=1,2,3) $$

is an algebra isomorphism $\mathrm{Cl}_{3,0}\to\mathbb{B}$: indeed $(ie_k)^2=i^2e_k^2=(-1)(-1)=+1$ and the images anticommute, while both algebras are eight-dimensional over $\mathbb{R}$ and isomorphic to $M_2(\mathbb{C})$. Under this identification the quaternion units themselves are the bivectors, with, in the quaternion convention $e_1e_2=e_3$ of the category,

$$ \gamma_2\gamma_3\longmapsto(ie_2)(ie_3)=-e_2e_3=-e_1, \qquad e_1=-\gamma_2\gamma_3, $$

and cyclically; the even subalgebra of $\mathrm{Cl}_{3,0}$ is a copy of $\mathbb{H}$:

$$ \mathrm{Cl}^0_{3,0}=\operatorname{span}\{1,\gamma_1\gamma_2,\gamma_2\gamma_3,\gamma_3\gamma_1\}\cong\mathbb{H}. $$

The volume element $\omega=\gamma_1\gamma_2\gamma_3$ maps to $i^3e_1e_2e_3=(-i)(-1)=i$, since $e_1e_2e_3=e_3e_3=-1$ in the quaternion convention of the category, so under the identification the volume element is the central scalar imaginary of $\mathbb{B}$.

The mixed cases. $\mathrm{Cl}_{1,2}$ and $\mathrm{Cl}_{2,1}$ are obtained by the same matrix method or from the graded tensor product:

$$ \mathrm{Cl}_{1,2}\cong\mathrm{Cl}_{0,2}\,\hat{\otimes}\,\mathrm{Cl}_{1,0}\cong\mathbb{H}\,\hat{\otimes}\,\mathbb{D}, \qquad \mathrm{Cl}_{2,1}\cong\mathrm{Cl}_{1,1}\,\hat{\otimes}\,\mathrm{Cl}_{1,0}. $$

The graded tensor product is not the ordinary tensor product of the underlying algebras: the Koszul sign makes $\mathbb{H}\,\hat{\otimes}\,\mathbb{D}$ simple, isomorphic to $M_2(\mathbb{C})$, whereas $\mathbb{H}\otimes_{\mathbb{R}}\mathbb{D}\cong\mathbb{H}\times\mathbb{H}$ has two factors. Computing the algebra directly, $\mathrm{Cl}_{1,2}\cong\mathrm{Cl}_{3,0}\cong M_2(F(\sqrt{-1}))$ by the identity $\mathrm{Cl}_{p+1,q}\cong\mathrm{Cl}_{q+1,p}$, its volume element having $\omega^2=(-1)^3(+1)(-1)(-1)=-1$ and hence center $F(\sqrt{-1})$. For $\mathrm{Cl}_{2,1}$ one has $\omega^2=(-1)^3(+1)(+1)(-1)=+1$, so the center is $F\times F$ and the algebra is split: $\mathrm{Cl}_{2,1}\cong M_2(F)\times M_2(F)$, in agreement with the graded tensor product $\mathrm{Cl}_{1,1}\hat\otimes\mathrm{Cl}_{1,0}\cong M_2(F)\hat\otimes(F\times F)$. The table below summarises the cases of dimension at most three.

Remark. The passage from $\mathrm{Cl}_{1,1}\cong M_2(F)$ to $\mathrm{Cl}_{1,2}\cong M_2(\mathbb{C})$ and from $\mathrm{Cl}_{2,0}\cong M_2(F)$ to $\mathrm{Cl}_{2,1}\cong M_2(F)\times M_2(F)$ shows that adjoining a generator of square $+1$ to a common real matrix algebra splits it into two factors, while adjoining a generator of square $-1$ complexifies it. This is the local form of the periodicity.

The Even Subalgebra Recursion

The even subalgebra of a Clifford algebra is again a Clifford algebra, with the rank reduced by one. This is the computational engine of the whole classification, and it is worth isolating in the low-dimensional cases where it can be seen directly.

Theorem. Let $V$ be a non-degenerate quadratic space over $F$ with an orthogonal basis $e_1,\dots,e_n$, and suppose that $q$ takes the value $-1$ on some generator. Then

$$ \mathrm{Cl}^0(V,q)\cong \mathrm{Cl}(V',q'), $$

where $V'$ has dimension $n-1$. More precisely, if $q(e_n)=-1$ then the linear map

$$ V'=e_n^{\perp}\longrightarrow \mathrm{Cl}^0(V,q), \qquad v\longmapsto e_n v $$

is an isometry onto a set of generators of the even subalgebra, and $\mathrm{Cl}^0(V,q)$ is generated by the $n-1$ elements $e_ne_1,\dots,e_ne_{n-1}$.

Proof. For $v\in e_n^\perp$ the product $e_nv$ is even. It satisfies

$$ (e_nv)(e_nv)=e_nv e_nv=-e_n^2v^2=v^2, $$

so the elements $e_nv$ square to $q(v)$, and for two vectors $u,v$ orthogonal to each other and to $e_n$,

$$ (e_nu)(e_nv)+(e_nv)(e_nu)=e_nu e_nv+e_nv e_nu=-e_n^2(uv+vu)=-(uv+vu)=0, $$

where the middle step uses that $u$ and $v$ both anticommute with $e_n$ and $e_n^2=-1$. The products $e_ne_i$ therefore satisfy the Clifford relations of the form $q$ restricted to $e_n^\perp$, and by the universal property they generate a quotient of $\mathrm{Cl}(e_n^\perp,q)$. Both sides have dimension $2^{n-1}$, so the map is an isomorphism.

Low-dimensional instances. For $\mathrm{Cl}_{0,2}$ one has $\mathrm{Cl}^0_{0,2}=\operatorname{span}\{1,e_1e_2\}\cong\mathbb{C}$; for $\mathrm{Cl}_{3,0}$ one has $\mathrm{Cl}^0_{3,0}\cong\mathbb{H}$, as computed above; and for $\mathrm{Cl}_{1,3}$ the recursion applied to a generator of square $-1$ gives $\mathrm{Cl}^0_{1,3}\cong\mathrm{Cl}_{1,2}$, which is $M_2(\mathbb{C})$ by the identification of the next section, the same algebra that is used in the biquaternion articles. The recursion holds whenever a generator of square $-1$ exists; when a generator of square $+1$ is used instead, the same computation returns $-q$ in place of $q$, which is why the sign of the form matters and why the recursion alternates between the two definite families.

Complexification

Passing from a real to a complex Clifford algebra removes the distinction between the two signs, because $-1$ becomes a square. Applied to the tensor product, this gives the complexification of a real Clifford algebra.

Proposition. Let $V$ be a real quadratic space. Then

$$ \mathrm{Cl}(V,q)\otimes_{\mathbb{R}}\mathbb{C}\cong \mathbb{C}\mathrm{l}(V\otimes_{\mathbb{R}}\mathbb{C}), $$

the complex Clifford algebra of the complexified form, and the complexification depends only on $n=\dim V$, not on the signature.

Proof. The complexification of the tensor algebra is $T(V)\otimes_{\mathbb{R}}\mathbb{C}\cong T(V\otimes_{\mathbb{R}}\mathbb{C})$, and the ideal of the Clifford relations is carried to the ideal of the relations $v^2=q(v)1$ over $\mathbb{C}$; the quotient is $\mathbb{C}\mathrm{l}(V\otimes_{\mathbb{R}}\mathbb{C})$. For the second statement, over $\mathbb{C}$ every non-degenerate form can be put in the form $z_1^2+\cdots+z_n^2$ by a linear change of coordinates, since $\pm1$ are both squares over $\mathbb{C}$; hence the complex Clifford algebra depends only on $n$.

The complex classification. Writing $\mathbb{C}\mathrm{l}_n$ for the complex Clifford algebra of an $n$-dimensional non-degenerate form, the complex case is periodic of period two:

$$ \mathbb{C}\mathrm{l}_{n+2}\cong \mathbb{C}\mathrm{l}_n\otimes_{\mathbb{C}} M_2(\mathbb{C}), $$

because the graded tensor product with the two-dimensional complex algebra is $M_2(\mathbb{C})$, and one has

$$ \mathbb{C}\mathrm{l}_{2m}\cong M_{2^m}(\mathbb{C}), \qquad \mathbb{C}\mathrm{l}_{2m+1}\cong M_{2^m}(\mathbb{C})\times M_{2^m}(\mathbb{C}). $$

For example $\mathbb{C}\mathrm{l}_1\cong\mathbb{C}\times\mathbb{C}$, $\mathbb{C}\mathrm{l}_2=M_2(\mathbb{C})$, $\mathbb{C}\mathrm{l}_3=M_2(\mathbb{C})\times M_2(\mathbb{C})$, $\mathbb{C}\mathrm{l}_4=M_4(\mathbb{C})$. This two-fold periodicity is the complex shadow of the eight-fold real periodicity; the difference between the two is the difference between $\mathbb{C}$ and $\mathbb{R}$ as the ground field, that , between the algebraic closure and the real form.

Realification

The converse operation associates to a complex Clifford algebra its real forms, and to a real Clifford algebra the complex algebra it complexifies to. Two constructions are needed.

Restriction of scalars. A complex vector space of dimension $m$ is a real vector space of dimension $2m$. Accordingly, if $A$ is a complex algebra of complex dimension $d$, then its underlying real algebra $A_{\mathbb{R}}$ has real dimension $2d$. For the complex matrix algebra this gives an underlying real algebra of real dimension $2m^{2}$, which for $m=2$ is the eight-dimensional real algebra $M_2(\mathbb{C})_{\mathbb{R}}$ of the biquaternion articles.

The real forms of $\mathbb{C}\mathrm{l}_n$. For each $n$ the complex algebra $\mathbb{C}\mathrm{l}_n$ has real forms, among them the real Clifford algebras $\mathrm{Cl}_{p,q}$ with $p+q=n$; these are not the only real forms, and it is the Clifford ones that the sign-sensitive classification enumerates. In the low-dimensional cases they are visible: $\mathbb{C}\mathrm{l}_2=M_2(\mathbb{C})$ has the Clifford real forms $M_2(\mathbb{R})$ and $\mathbb{H}$, which are $\mathrm{Cl}_{2,0}\cong\mathrm{Cl}_{1,1}$ and $\mathrm{Cl}_{0,2}$ respectively; and $\mathbb{C}\mathrm{l}_3=M_2(\mathbb{C})\times M_2(\mathbb{C})$ has the Clifford real forms $M_2(\mathbb{C})$, $M_2(\mathbb{R})\times M_2(\mathbb{R})$ and $\mathbb{H}\times\mathbb{H}$, which are $\mathrm{Cl}_{3,0}\cong\mathrm{Cl}_{1,2}$, $\mathrm{Cl}_{2,1}$ and $\mathrm{Cl}_{0,3}$. Realification is therefore not a single operation but a passage from a complex algebra to its table of real forms, and the table is exactly the subject of the real classification.

The relation between the two periodicity phenomena. The complex classification is periodic of period two because $\mathbb{C}\mathrm{l}_{n+2}\cong\mathbb{C}\mathrm{l}_n\otimes M_2(\mathbb{C})$. The real classification is periodic of period eight. The two are not independent: the real periodicity reduces modulo the complex one, and the eight real classes fall into the two complex classes according to the parity of $n$. The precise statement is not covered here.

The Table

The following table collects the cases of dimension at most three over a field $F$ of characteristic not $2$. The entries name the algebra up to isomorphism; $M_m(F)$ is the algebra of $m\times m$ matrices.

The table is stated for a field of characteristic not $2$ in which $-1$ is not a square and in which the quaternion algebra $(-1,-1)_F$ is a division algebra; this is the case $F=\mathbb{R}$. Over a general field of characteristic not $2$ the entries $F(\sqrt{-1})$, $\mathbb{H}$ and $\mathbb{H}\times\mathbb{H}$ are replaced by the corresponding split forms, each under its own hypothesis: $F(\sqrt{-1})\cong F\times F$ when $-1$ is a square in $F$, and the quaternion algebra $(-1,-1)_F\cong M_2(F)$ when it splits, as it does over every finite field of odd characteristic. The two hypotheses are independent: over a finite field the quaternion algebra always splits, whether or not $-1$ is a square there. The mixed entries are governed by the volume element, whose square is the discriminant of the center: the center of $\mathrm{Cl}_{3,0}$ and of $\mathrm{Cl}_{1,2}$ is $F(\sqrt{-1})$, so these become $M_2(F)\times M_2(F)$ exactly when $-1$ is a square in $F$, while $\mathrm{Cl}_{0,3}$ is the product $(-1,-1)_F\times(-1,-1)_F$, with center $F\times F$ in every case, and becomes $M_2(F)\times M_2(F)$ exactly when the quaternion algebra splits. The resulting tables over such a field are read from the general classification.

Echoing the one-dimensional case, one has $\mathrm{Cl}_{1,0}\cong\mathbb{D}$ and $\mathrm{Cl}_{0,1}\cong\mathbb{C}$ when $F=\mathbb{R}$; $\mathrm{Cl}_{3,0}\cong\mathbb{C}\mathrm{l}_2\cong M_2(\mathbb{C})\cong\mathbb{B}$, the biquaternion algebra; $\mathrm{Cl}_{1,1}$ is the four-dimensional algebra called the split quaternions in the classical literature; and $\mathrm{Cl}_{0,2}\cong\mathbb{H}$ is the quaternion algebra. The entries with center larger than $F$ are exactly those of odd rank $n$, in agreement with the general rule that the volume element is central and nontrivial when $n$ is odd.

Summary

The Clifford algebras of small non-degenerate quadratic forms are computed directly. In one dimension the sign of the form decides everything: $\mathrm{Cl}_{1,0}\cong F\times F=\mathbb{D}$ and $\mathrm{Cl}_{0,1}\cong F(\sqrt{-1})$. In two dimensions the negative definite form gives the quaternion algebra $\mathrm{Cl}_{0,2}\cong\mathbb{H}$, while $\mathrm{Cl}_{2,0}\cong\mathrm{Cl}_{1,1}\cong M_2(F)$ are split and isomorphic to one another. In three dimensions $\mathrm{Cl}_{3,0}\cong M_2(F(\sqrt{-1}))$ is the complex $2\times2$ matrix algebra, identified with the biquaternion algebra $\mathbb{B}$ through the generators $\gamma_k\mapsto ie_k$; $\mathrm{Cl}_{1,2}$ is the same matrix algebra; and the two algebras with a central volume element of square $+1$, namely $\mathrm{Cl}_{0,3}\cong\mathbb{H}\times\mathbb{H}$ and $\mathrm{Cl}_{2,1}\cong M_2(F)\times M_2(F)$, split into two factors.

The even subalgebra of $\mathrm{Cl}(V,q)$ is again a Clifford algebra, with rank reduced by one, whenever a generator of square $-1$ is available; this recursion is the computational origin of the classification and is the reason the pattern is periodic rather than merely multiplicative.

Complexification sends $\mathrm{Cl}(V,q)$ to the complex Clifford algebra of the complexified form and forgets the signature; the complex classification is periodic of period two, with $\mathbb{C}\mathrm{l}_{2m}\cong M_{2^m}(\mathbb{C})$ and $\mathbb{C}\mathrm{l}_{2m+1}\cong M_{2^m}(\mathbb{C})\times M_{2^m}(\mathbb{C})$. Realification is the passage from a complex algebra to its table of real forms, and the table is the real classification.

Summary of Notation

Further Reading

• H. Blaine Lawson and Marie-Louise Michelsohn, Spin Geometry (Princeton University Press, 1989), for the classification of real and complex Clifford algebras and the low-dimensional computations.
• Ian R. Porteous, Clifford Algebras and the Classical Groups (Cambridge University Press, 1995), for the relation between the small Clifford algebras and the classical groups.
• Pertti Lounesto, Clifford Algebras and Spinors (Cambridge University Press, 2nd ed. 2001), for explicit low-dimensional identifications and the Pauli-matrix model.
• T. Y. Lam, Introduction to Quadratic Forms over Fields (American Mathematical Society, 2005), for the algebra of quadratic spaces and the role of the discriminant in the low-dimensional classification.
• Max-Albert Knus, Quadratic and Hermitian Forms over Rings (Springer, 1991), for the behaviour of the classification under base change.