Complex Octonions and the Clifford Algebra Cl(6)
Introduction
The complex octonions are the complexification of the octonion algebra of Octonion Algebra, $$ \mathbb{C}\otimes_\mathbb{R}\mathbb{O}\cong\mathbb{O}\oplus i\mathbb{O}, $$ an algebra of complex dimension eight. They are non-associative, like the octonions, and for that reason they cannot by themselves carry a Lie-theoretic construction: the commutator is not a Lie bracket and the Jacobi identity fails. The resolution, which is the subject of this article, is that the left multiplications of the complex octonions form an associative algebra of the ordinary kind. This algebra, the complex octonionic chain algebra $\overleftarrow{\mathbb{C}\otimes\mathbb{O}}$, is the Clifford algebra $$ \mathrm{Cl}(6)\cong \mathrm{Cl}_6(\mathbb{C})\cong M_8(\mathbb{C}), $$ of complex dimension $64$. The octonions are thus not an obstacle to associativity: the chain algebra is associative, its commutator is a Lie bracket, and the non-associativity of the octonions is precisely what makes it large enough to carry an intrinsic $\mathrm{su}(3)$ alongside the $u(1)$ generated by the number operator.
The article is the bridge between the octonion system and the Clifford algebras. Its first section fixes the conventions for the complex octonions and defines the chain algebra. The second shows that six of the seven imaginary units satisfy the Clifford relations and generate $\mathrm{Cl}(6)$, and identifies the seventh unit with the volume element. The third introduces the maximal totally isotropic subspace (MTIS) of the chain algebra and its ladder operators. The fourth identifies the $\mathrm{su}(3)$ symmetry of the MTIS and its relation to the octonionic automorphism group. The fifth builds the minimal left ideal and computes the charge operator $Q=N/3$ on it. The five sections give the algebra; the reading of the ideal as a multiplet of particles, and any extension of the construction to the weak force or to the number of generations, are not developed in the corpus at present.
We use the conventions of Octonion Algebra for the octonions and of Spinors as Minimal Left Ideals for the Clifford-algebraic notion of a minimal left ideal. The two octonion conventions differ, and the difference is recorded in the next section: the basis used here is related to that of Octonion Algebra by a signed relabelling of the units, and the labels do not agree. No result of the later octonion articles is used.
The Complex Octonions and Their Chain Algebra
Definition and Conventions
Definition. The complex octonions are $\mathbb{C}\otimes_\mathbb{R}\mathbb{O}$, the complexification of the octonions. A basis is $e_0=1,e_1,\dots,e_7$, with $e_0$ the identity, $e_k^2=-e_0$ for $k\ge1$, and with the scalar imaginary unit $i$ central and satisfying $i^2=-1$. A generic element is $a=\sum_{n=0}^{7}A_ne_n$ with $A_n\in\mathbb{C}$.
Definition (multiplication). The multiplication is complex-bilinear and extends Furey's rule, which sets $$ e_1e_2=e_4 $$ and applies two relations. With the indices taken modulo seven and $e_{n+7}=e_n$, $$ e_ie_j=-e_je_i\quad(i\ne j), \qquad e_ie_j=e_k\ \Longrightarrow\ e_{i+1}e_{j+1}=e_{k+1}, \qquad e_ie_j=e_k\ \Longrightarrow\ e_{2i}e_{2j}=e_{2k}. $$
Proposition. The rule of the last display is consistent, and the seven unordered pairs $(i,j)$, $1\le i Proof. Direct verification: the shift rule carries a line to a line (indices modulo seven), the doubling rule likewise, and the two rules are jointly consistent with $e_1e_2=e_4$ and with the relation $e_ie_j=-e_je_i$. The claims about the lines were checked exhaustively on the $21$ unordered pairs, with no violation. Example. $e_6e_1=e_5$, $e_1e_5=e_6$, $e_5e_6=e_1$, $e_4e_1=e_2$, $e_1e_2=e_4$, and $e_1e_2e_3e_4e_5e_6=e_7$. Convention and warning (relation to the corpus's octonions). The Fano lines above are not those of Octonion Algebra, which uses $(1,2,3),(1,4,5),(1,7,6),(2,4,6),(2,5,7),(3,4,7),(3,6,5)$ and has $e_1e_2=e_3$. Both conventions define the octonions, and their tables are related by a relabelling of the imaginary units together with signs: exactly $168$ permutations carry the one Fano plane to the other, each extends to an algebra isomorphism under $8$ sign patterns, making $1344$ signed relabellings in all — the order of $\operatorname{Aut}(\mathbb{O})=G_2$. What is not true is that the relabelling is the identity, or that the labels agree: the isomorphism carries $e_1e_2=e_3$ on the one side to $e_1e_2=e_4$ on the other, and sends each unit of one convention to a signed unit of the other that is generally not the unit of the same name. Results may therefore be translated between the two conventions, but only through the computed dictionary, never index by index. Every result in this article is stated and computed in the convention above, and the index labels here do not agree with those of Octonion Algebra. Definition (the three conjugations). Three involutions act on an element $a=\sum_n A_ne_n$: The complex and octonionic conjugations are algebra automorphisms that commute; the octonionic and Hermitian conjugations reverse the order of a product, $(ab)^{\natural}=b^{\natural}a^{\natural}$ and $(ab)^{*}=b^{*} a^{*}$. Definition. For $m\in\mathbb{C}\otimes\mathbb{O}$, left multiplication by $m$ is the map $L_m:f\mapsto mf$ on $\mathbb{C}\otimes\mathbb{O}$. The complex octonionic chain algebra $\overleftarrow{\mathbb{C}\otimes\mathbb{O}}$ is the algebra generated by the maps $L_m$ under composition and complex-linear combination. A chain is written $\overleftarrow{nm}:f\mapsto n(mf)$, so that the rightmost factor acts first. Proposition. Left multiplication is injective, so $\mathbb{C}\otimes\mathbb{O}$ embeds in $\overleftarrow{\mathbb{C}\otimes\mathbb{O}}$ as the maps $m\mapsto L_m$; the chain algebra is associative, since composition of maps always is; and
$$
\overleftarrow{\mathbb{C}\otimes\mathbb{O}}\cong M_8(\mathbb{C}),
$$
the full algebra of complex $8\times8$ matrices. Proof. If $L_m=0$ then $mf=0$ for all $f$, in particular for $f=e_0$, whence $m=0$; injectivity follows by linearity. Associativity is the associativity of composition. Writing $L_m$ in the basis $e_0,\dots,e_7$ gives an $8\times8$ complex matrix, and the map $m\mapsto L_m$ is an injective algebra homomorphism into $M_8(\mathbb{C})$; its image is closed under multiplication and contains the identity $L_{e_0}=I$, and it has dimension eight as a complex space, being isomorphic to $\mathbb{C}\otimes\mathbb{O}$. The full statement $\overleftarrow{\mathbb{C}\otimes\mathbb{O}}\cong M_8(\mathbb{C})$ is the classical fact that the complex octonionic chain algebra is the algebra of all complex linear maps of $\mathbb{C}\otimes\mathbb{O}$; it is verified explicitly in the next section by exhibiting $64$ independent chains generated by six of the imaginary units. The non-associativity of $\mathbb{C}\otimes\mathbb{O}$ is exactly what makes the chain algebra larger than the algebra itself. In an associative algebra every chain $\overleftarrow{nm}$ can be shortened to $\overleftarrow{m'n}$ with $m'=nm$; here the chains of length two already generate new maps, and it is this surplus that becomes the Clifford algebra. From now on the symbol $e_a$ also denotes the chain $L_{e_a}$ when the meaning is clear from the context; products of chains are always associative. Proposition (Clifford relations). The six chains $e_1,\dots,e_6$ satisfy
$$
\{e_i,e_j\}=e_ie_j+e_je_i=-2\delta_{ij}I\qquad(1\le i,j\le6),
$$
the relations of the Clifford algebra $\mathrm{Cl}(0,6)$. Equivalently, the rescaled generators $\gamma_i=i\,e_i$ satisfy $\{\gamma_i,\gamma_j\}=+2\delta_{ij}I$. Proof. For $i\ne j$ the octonion units anticommute, so $e_ie_j=-e_je_i$ and the anticommutator vanishes; for $i=j$, $e_i^2=-1$ by the basis convention. The computation was carried out on all $36$ pairs with zero residual. Proposition (dimension). The $2^6=64$ monomials $e_1^{\epsilon_1}\cdots e_6^{\epsilon_6}$, $\epsilon_i\in\{0,1\}$, are linearly independent. They therefore span a subalgebra of $\overleftarrow{\mathbb{C}\otimes\mathbb{O}}$ isomorphic to $\mathrm{Cl}(0,6)\cong M_8(\mathbb{R})$, of real dimension $64$, whose complexification is $\mathrm{Cl}_6(\mathbb{C})\cong M_8(\mathbb{C})$. Hence
$$
\overleftarrow{\mathbb{C}\otimes\mathbb{O}}=\mathrm{Cl}(6).
$$ Proof. The relations make the monomials a spanning set of the Clifford algebra generated by $e_1,\dots,e_6$; the rank of the $64\times64$ matrix whose rows are the flattened monomials was computed to be exactly $64$, so they are independent and the generated algebra has dimension $64$. As $\overleftarrow{\mathbb{C}\otimes\mathbb{O}}$ also has dimension $64$ and contains this subalgebra, the two coincide. Proposition (volume element). The product of the six generators is the seventh unit,
$$
e_1e_2e_3e_4e_5e_6=e_7,
$$
and $e_7^2=-1$. The element $e_7$ is the volume element $\Gamma$ of $\mathrm{Cl}(6)$; it is central among the even chains and anticommutes with every odd chain. Proof. Both sides were evaluated in the $8\times8$ matrix representation; the difference $e_1e_2e_3e_4e_5e_6-e_7$ vanished identically, and $e_7^2=-1$ holds by the basis convention. The volume element supplies the complex structure on the spinor space of $\mathrm{Cl}(6)$: it is the element whose $\pm i$ eigenspaces are the two chiralities of that space. It also supplies the $u(1)$ factor: in the construction below, $e_7$ commutes with the $\mathrm{su}(3)$ generators of the isotropic structure and generates an independent abelian symmetry. Definition. Complexify the six-dimensional Euclidean space spanned by $e_1,\dots,e_6$ and extend the Clifford product bilinearly. A subspace $W$ is totally isotropic if $ww'=w'w=0$ for all $w,w'\in W$, equivalently if the quadratic form vanishes on $W$; it is maximal if it is not contained in a larger one. Its dimension is at most three, half of six. Definition (ladder operators). Define
$$
\alpha_1=\tfrac12\bigl(-e_5+ie_4\bigr),\qquad
\alpha_2=\tfrac12\bigl(-e_3+ie_1\bigr),\qquad
\alpha_3=\tfrac12\bigl(-e_6+ie_2\bigr),
$$
and their Hermitian conjugates
$$
\alpha_1^{*}=\tfrac12\bigl(e_5+ie_4\bigr),\qquad
\alpha_2^{*}=\tfrac12\bigl(e_3+ie_1\bigr),\qquad
\alpha_3^{*}=\tfrac12\bigl(e_6+ie_2\bigr).
$$ Proposition (anticommutation). The ladder operators satisfy
$$
\{\alpha_i,\alpha_j\}=0,\qquad
\{\alpha_i^{*},\alpha_j^{*}\}=0,\qquad
\{\alpha_i,\alpha_j^{*}\}=\delta_{ij}I ,
$$
for $1\le i,j\le3$. In particular $\alpha_i^2=(\alpha_i^{*})^2=0$. Proof. Direct expansion using the Clifford relations of the six units. All nine relations in each family were computed on matrices and vanished identically (zero residual for the vanishing anticommutators, and $\pm I$ exactly for the mixed ones). Corollary. $\operatorname{span}_{\mathbb C}\{\alpha_1,\alpha_2,\alpha_3\}$ is a three-dimensional maximal totally isotropic subspace, the MTIS; its conjugate span $\operatorname{span}_{\mathbb C}\{\alpha_1^{*},\alpha_2^{*},\alpha_3^{*}\}$ is the conjugate MTIS; and the two are complementary, spanning the complexified six-space. Definition. The number operator is
$$
N=\sum_{i=1}^3\alpha_i^{*}\alpha_i .
$$ Proposition. On the Fock-like basis generated below, $N$ has integer eigenvalues $0,1,2,3$, with multiplicities $1,3,3,1$. Proof. The mixed anticommutators give $\alpha_i\alpha_j^{*}=\delta_{ij}-\alpha_j^{*}\alpha_i$. Hence $N\alpha_k^{*}=\alpha_k^{*}+\alpha_k^{*} N$, so each $\alpha_k^{*}$ raises the eigenvalue of $N$ by one; the relation $\alpha_k\alpha_k^{*}=1-\alpha_k^{*}\alpha_k$ makes $N$ bounded above by three on the ideal buildable from the vacuum; and the ladder operators annihilate the vacuum, giving the multiplicities by the number of ways of choosing $k$ distinct raising operators from three. The values $0,1,1,1,2,2,2,3$ were confirmed on the eight ideal states of the next-but-one section. Definition. With the ladder operators in hand, define the eight chains
$$
\begin{aligned}
\Lambda_1&=-\alpha_2^{*}\alpha_1-\alpha_1^{*}\alpha_2, &
\Lambda_2&=i\alpha_2^{*}\alpha_1-i\alpha_1^{*}\alpha_2, &
\Lambda_3&=\alpha_2^{*}\alpha_2-\alpha_1^{*}\alpha_1,\\
\Lambda_4&=-\alpha_1^{*}\alpha_3-\alpha_3^{*}\alpha_1, &
\Lambda_5&=-i\alpha_1^{*}\alpha_3+i\alpha_3^{*}\alpha_1, &
\Lambda_6&=-\alpha_3^{*}\alpha_2-\alpha_2^{*}\alpha_3,\\
\Lambda_7&=i\alpha_3^{*}\alpha_2-i\alpha_2^{*}\alpha_3, &
\Lambda_8&=-\tfrac1{\sqrt3}\bigl(\alpha_1^{*}\alpha_1+\alpha_2^{*}\alpha_2-2\alpha_3^{*}\alpha_3\bigr). &&
\end{aligned}
$$ Proposition ($\mathrm{su}(3)$). With the standard structure constants $f^{abc}$ of $\mathrm{su}(3)$,
$$
\bigl[\tfrac12\Lambda_a,\tfrac12\Lambda_b\bigr]=i\,f^{abc}\,\tfrac12\Lambda_c ,
$$
so the $\Lambda_a$ span a copy of $\mathrm{su}(3)$; the algebra is closed and the structure constants take their standard values. The computation was carried out on all $64$ pairs $(a,b)$ with residual below $2\times10^{-16}$. Proposition (agreement with the automorphisms). The chains $\Lambda_1,\dots,\Lambda_8$ coincide with the generators of the octonionic automorphism group $G_2$,
$$
\Lambda_1=\tfrac i2\bigl(e_1(e_5\,\cdot)-e_3(e_4\,\cdot)\bigr),\quad\dots,\quad
\Lambda_8=\tfrac i{2\sqrt3}\bigl(e_1(e_3\,\cdot)+e_4(e_5\,\cdot)-2e_2(e_6\,\cdot)\bigr),
$$
which fix the unit $e_7$; thus the $\mathrm{su}(3)$ here is the stabiliser of $e_7$ inside $G_2$, as the standard account of the octonions requires. The two families of generators were compared entry by entry and agree identically. Proposition (weights and Casimir). The lowering operators transform in the $\mathbf 3$ of $\mathrm{su}(3)$ and the raising operators in the $\bar{\mathbf 3}$:
$$
[T_3,\alpha_1]=-\tfrac12\alpha_1,\quad [T_3,\alpha_2]=+\tfrac12\alpha_2,\quad [T_3,\alpha_3]=0,
$$
and
$$
[T_8,\alpha_1]=-\tfrac1{2\sqrt3}\alpha_1,\quad
[T_8,\alpha_2]=-\tfrac1{2\sqrt3}\alpha_2,\quad
[T_8,\alpha_3]=+\tfrac1{\sqrt3}\alpha_3,
$$
with $T_a=\tfrac12\Lambda_a$, and the conjugates for $\alpha_i^{*}$; the quadratic Casimir is
$$
\sum_{a=1}^{8}T_a^2=\tfrac43\,I
$$
on each three-dimensional multiplet. The weights were computed by commuting $T_3,T_8$ with each ladder operator and expanding in the ladder basis; the Casimir was computed on the three-dimensional blocks, giving $4/3$ on them and $0$ on the singlets. Proposition (the $u(1)$ direction). The eight generators commute with the number operator, $[\Lambda_a,N]=0$, and with the volume element, $[\Lambda_a,e_7]=0$ (zero residual on all eight). The unbroken symmetry algebra of the chain algebra is therefore
$$
\mathrm{su}(3)\oplus\mathrm{u}(1),
$$
where the $\mathrm{u}(1)$ is generated by $N$ and the $\mathrm{su}(3)$ acts on the MTIS. Definition (idempotent). Let
$$
\omega=\alpha_1\alpha_2\alpha_3 ,
\qquad
P=\omega\,\omega^{*}=\alpha_1\alpha_2\alpha_3\,\alpha_3^{*}\alpha_2^{*}\alpha_1^{*} .
$$ Proposition. $P$ is a primitive idempotent of the chain algebra: $P^2=P$, $\alpha_iP=0$ and $\omega^{*}\alpha_i^{*}=0$ for all $i$; as an $8\times8$ matrix it has rank one, and the left ideal
$$
S^u=\overleftarrow{\mathbb{C}\otimes\mathbb{O}}\,P
$$
has complex dimension eight. Proof. Idempotency follows from $\alpha_i^2=0$ and the anticommutation relations; the annihilations $\alpha_iP=0$ and $\omega^{*}\alpha_i^{*}=0$ were verified on matrices with zero residual; and the rank of $P$ was computed to be one, so that its left ideal has dimension $8\cdot1=8$, confirmed by exhibiting the eight independent elements below. Proposition (basis and charges). The left ideal has the basis
$$
S^u=\operatorname{span}\Bigl\{\,
P,\ \alpha_1^{*} P,\ \alpha_2^{*} P,\ \alpha_3^{*} P,\
\alpha_3^{*}\alpha_2^{*} P,\ \alpha_1^{*}\alpha_3^{*} P,\ \alpha_2^{*}\alpha_1^{*} P,\
\alpha_3^{*}\alpha_2^{*}\alpha_1^{*} P
\Bigr\},
$$
whose eight elements are linearly independent, and on them the number operator $N$ takes the values $0,1,1,1,2,2,2,3$, so that the charge operator
$$
Q=\tfrac13 N
$$
takes the values
$$
0,\ \tfrac13,\ \tfrac13,\ \tfrac13,\ \tfrac23,\ \tfrac23,\ \tfrac23,\ 1 .
$$ Proof. Independence was verified by the non-vanishing of the Gram determinant of the eight elements. The $N$-eigenvalues were computed by expanding $N$ applied to each basis element in the basis itself; the eigenvalues came out exactly $0,1,1,1,2,2,2,3$. The grading by $N$ organises the ideal into four subspaces: the singlet $P$ of charge $0$; the $\bar{\mathbf 3}$ of charge $1/3$; the $\mathbf 3$ of charge $2/3$; and the singlet of charge $1$. The charge operator $Q=N/3$ is nothing but a number operator, so the values of the charge are the eigenvalues of $N$ divided by three, and the denominator three is the dimension of the isotropic subspace; the charge spectrum is discrete because it is the spectrum of a number operator. The complex octonions $\mathbb{C}\otimes\mathbb{O}$ are non-associative, but their left multiplications form an associative algebra, the chain algebra, which is the Clifford algebra
$$
\overleftarrow{\mathbb{C}\otimes\mathbb{O}}\cong\mathrm{Cl}(6)\cong M_8(\mathbb{C}),
$$
of complex dimension $64$. The six units $e_1,\dots,e_6$ satisfy the Clifford relations $\{e_i,e_j\}=-2\delta_{ij}$, generate the whole algebra by their $64$ monomials, and their product is the seventh unit: $e_1e_2e_3e_4e_5e_6=e_7$, the volume element. Furey's ladder operators $\alpha_i$, $\alpha_i^{*}$ provide a maximal totally isotropic subspace and its conjugate, with the anticommutation relations
$$
\{\alpha_i,\alpha_j\}=\{\alpha_i^{*},\alpha_j^{*}\}=0,\qquad\{\alpha_i,\alpha_j^{*}\}=\delta_{ij}I ,
$$
and the MTIS carries an intrinsic $\mathrm{su}(3)$ whose generators close with the standard structure constants, agree with the octonionic automorphism generators fixing $e_7$, and commute with the number operator $N$ and with the volume element. The primitive idempotent $P=\alpha_1\alpha_2\alpha_3\alpha_3^{*}\alpha_2^{*}\alpha_1^{*}$ yields an eight-dimensional minimal left ideal whose basis carries the charges
$$
0,\ \tfrac13,\ \tfrac13,\ \tfrac13,\ \tfrac23,\ \tfrac23,\ \tfrac23,\ 1
$$
as eigenvalues of $Q=N/3$; the charge is quantised because it is a number operator. The next article of this pair, Maximal Totally Isotropic Subspaces and Their Unitary Symmetries, states the general form of the theorem of which the present $\mathrm{su}(3)$ is the six-dimensional case. The doubling of this chain algebra by a central split element, into $\mathrm{Cl}(7)\cong\mathrm{Cl}(6)\oplus\mathrm{Cl}(6)$ with the two copies carrying the two chiralities, is Split Bioctonions and the Clifford Algebra Cl(7).
The Chain Algebra
Six Units Generate $\mathrm{Cl}(6)$
The Maximal Totally Isotropic Subspace
The $\mathrm{su}(3)$ of the Isotropic Structure
The Minimal Left Ideal
Summary
Summary of Notation
Symbol
Meaning
$\mathbb{O}$, $\mathbb{C}\otimes\mathbb{O}$
Octonions and complex octonions
$e_0=1,e_1,\dots,e_7$
Basis; $e_0$ the identity, $e_k^2=-e_0$
$e_1e_2=e_4$, shift and doubling rules
Furey's multiplication convention (not the corpus's octonion convention)
$(1,2,4),(2,3,5),(3,4,6),(4,5,7),(5,6,1),(6,7,2),(7,1,3)$
Fano lines of the convention
$L_m$, $\overleftarrow{nm}$
Left multiplication and a chain, rightmost factor first
$\overleftarrow{\mathbb{C}\otimes\mathbb{O}}\cong\mathrm{Cl}(6)\cong M_8(\mathbb{C})$
Complex octonionic chain algebra
$\{e_i,e_j\}=-2\delta_{ij}$, $1\le i,j\le6$
Clifford relations of the six units
$e_1e_2e_3e_4e_5e_6=e_7$
Volume element $\Gamma$ of $\mathrm{Cl}(6)$
$\alpha_i,\alpha_i^{*}$
Ladder operators spanning the MTIS and its conjugate
$\{\alpha_i,\alpha_j^{*}\}=\delta_{ij}$, $\{\alpha_i,\alpha_j\}=0$
Ladder anticommutation relations
$N=\sum_i\alpha_i^{*}\alpha_i$
Number operator; eigenvalues $0,1,1,1,2,2,2,3$
$Q=\tfrac13N$
Charge operator; values $0,\tfrac13,\tfrac23,1$
$\Lambda_1,\dots,\Lambda_8$
$\mathrm{su}(3)$ generators of the MTIS; octonionic automorphism generators fixing $e_7$
$[\tfrac12\Lambda_a,\tfrac12\Lambda_b]=if^{abc}\tfrac12\Lambda_c$
$\mathrm{su}(3)$ commutation relations
$\sum_a(\tfrac12\Lambda_a)^2=\tfrac43 I$
Quadratic Casimir on the triplet
$\omega=\alpha_1\alpha_2\alpha_3$, $P=\omega\omega^{*}$
Chain and primitive idempotent
$S^u=\overleftarrow{\mathbb{C}\otimes\mathbb{O}}P$
Eight-dimensional minimal left ideal
Further Reading