Exceptional structures



An octonion or Cayley number is a kind of number similar to a quaternion number but with seven instead of just three square roots of unity adjoined, and satisfying certain relations.

The octonions, usually denoted 𝕆\mathbb{O}, form the largest of the four normed division algebras over the real numbers.



The octonions 𝕆\mathbb{O} is the nonassociative algebra over the real numbers which is generated from seven generators {e 1,,e 7}\{e_1, \cdots, e_7\} subject to the relations

  1. for all ii

    e i 2=1e_i^2 = -1

  2. for e ie je ke_i \to e_j \to e_k an edge or circle in the following diagram (a labeled version of the Fano plane) the relations

    1. e ie j=e ke_i e_j = e_k

    2. e je i=e ke_j e_i = -e_k

octonion multiplication table

(graphics grabbed from Baez 02)

This becomes a star-algebra with star involution

(1)()¯:𝕆𝕆 \overline{(-)} \;\colon\; \mathbb{O} \longrightarrow \mathbb{O}

which is the antihomomorphism ab¯=b¯a¯\overline{a b} = \overline{b} \overline{a} that is given on the above generators by

e i¯e iAAAAi{1,,7}. \overline{e_i} \coloneqq - e_i \phantom{AAAA} i \in \{1, \cdots, 7\} \,.

The product of all the generators with each other, bracketed to the right, is

e 7(e 6(e 5(e 4(e 3(e 2(e 11))))))=+1 e_7 (e_6 (e_5 (e_4 (e_3 ( e_2 (e_1 1 )))))) \;=\; + 1

By iteratively using the multiplication table in def. we compute as follows:

e 7(e 6(e 5(e 4(e 3(e 2e 1e 4))))) =e 7(e 6(e 5(e 4(e 3e 4e 6)))) =e 7(e 6(e 5(e 4e 6e 3))) =e 7(e 6(e 5e 3e 2)) =+e 7(e 6e 2e 7) =e 7e 7=1 =+1 \begin{aligned} & e_7 (e_6 (e_5 (e_4 (e_3 (\underset{-e_4}{\underbrace{e_2 e_1}}))))) \\ & = - e_7 (e_6 (e_5 (e_4 (\underset{e_6}{\underbrace{e_3 e_4}})))) \\ & = - e_7 (e_6 (e_5 (\underset{e_3}{\underbrace{e_4 e_6}}))) \\ & = - e_7 (e_6 (\underset{-e_2}{\underbrace{e_5 e_3}})) \\ & = + e_7 (\underset{-e_7}{\underbrace{e_6 e_2}}) \\ & = - \underset{= -1}{\underbrace{e_7 e_7}} \\ & = + 1 \end{aligned}

(real and imaginary octonions)

As for the complex numbers one says that

  • an imaginary octonion is an a𝕆a \in \mathbb{O} shuch that under the star involution (1) it is sent to its negative:

    a¯=a \overline{a} = -a
  • a real octonions is an a𝒪a \in \mathcal{O} shuch that under the star involution (1) it is sent to itself

    a¯=a \overline{a} = a

Accordingly every octonion decomposes into a real part and an imaginary part:

Re(a)12(a+a¯)AAIm(a)12(aa¯). Re(a) \coloneqq \tfrac{1}{2}(a + \overline{a}) \phantom{AA} Im(a) \coloneqq \tfrac{1}{2}(a - \overline{a}) \,.




The octonions are not an associative algebra. But the non-zero octonions and the unit octonions form Moufang loops.


(octonions are alternative)

The octonions form an alternative algebra


By linearity it is sufficient to check this on generators. So let e ie je ke_i \to e_j \to e_k be a circle or a cyclic permutation of an edge in the Fano plane as in Def. . Then by definition of the octonion multiplication we have

(e ie j)e j =e ke j =e je k =e i =e i(e je j) \begin{aligned} (e_i e_j) e_j &= e_k e_j \\ &= - e_j e_k \\ & = -e_i \\ & = e_i (e_j e_j) \end{aligned}

and similarly

(e ie i)e j =e j =e ke i =e ie k =e i(e ie j). \begin{aligned} (e_i e_i ) e_j &= - e_j \\ &= - e_k e_i \\ &= e_i e_k \\ &= e_i (e_i e_j) \end{aligned} \,.



The automorphism group of the octonions, as a real algebra, is the exceptional Lie group G2.

See also at normed division algebra – automorphism

Left multiplication by imaginary octonions


Given any octonion oo, then the operation of left multiplication by oo

𝕆 L o 𝕆 a oa \array{ \mathbb{O} &\overset{L_o}{\longrightarrow}& \mathbb{O} \\ a &\mapsto& o a }

is an \mathbb{R}-linear map. Under composition of linear maps, this defines an associative monoid acting linearly on 𝕆\mathbb{O}.


(Clifford action of imaginary octonions)

Consider the Clifford algebra

Cl(Im(𝕆),|| 2) Cl(Im(\mathbb{O}), -{\vert -\vert}^2)

on the underlying real vector space of that of the imaginary octonions (Def. ) regarded as an inner product space via the quadratic form given by the negative square norm.

Then the operation of left multiplication on 𝕆\mathbb{O} (def. ) induces a representation of this Clifford algebra on 8 𝕆\mathbb{R}^8 \simeq_{\mathbb{R}} \mathbb{O}.


By alternativity (Prop. ) we have for every vIm(𝕆)v \in Im(\mathbb{O}) and every a𝕆a \in \mathbb{O}

L vL v(a) v(va) =(vv)a =|v| 2a =L |v| 2(a) \begin{aligned} L_v L_v (a) & \coloneqq v (v a) \\ & = (v v) a \\ & = - {\vert v\vert}^2 a \\ & = L_{- {\vert v\vert}^2} (a) \end{aligned}

(consecutive left action by imaginary generators is unity)

The consecutive left multiplication action (Def. ) by all the imaginary octonion generators e ie_i (Def. ) is ±\pm the identity function on the octonions. Specifically, if one acts in increasing order of the labels in Def. , then it is +1:

L e 7L e 6L e 5L e 4L e 3L e 2L e 1=+Id 𝕆 L_{e_7} L_{e_6} L_{e_5} L_{e_4} L_{e_3} L_{e_2} L_{e_1} \;=\; + Id_{\mathbb{O}}

All the generators e ie_i are imaginary octonions (Def. ). By Prop. their left action on 𝕆\mathbb{O} represents a Clifford algebra-action of Cl(Im(𝕆),|| 2)Cl 0,7Cl(Im(\mathbb{O}), -{\vert-\vert}^2) \simeq Cl_{0,7} on 8 𝕆\mathbb{R}^{8} \simeq_{\mathbb{R}} \mathbb{O}.

By the classification of real Clifford algebras, Cl 0,7Cl_{0,7} has, up to isomorphism, two different irreducible modules. Their underlying vector space is 8\mathbb{R}^8 in both cases, and so the left action of imaginary octonions we have must be one of the two. The two irreps may be distinguished by the action of the “volume element” Γ 7Γ 6Γ 1\Gamma_7 \Gamma_6 \cdots \Gamma_1: On one of the two it acts as the identity, on the other as minus the identity.

Hence we may check the remaining sign by acting on any one octonion, for instance on the unit 1𝕆1 \in \mathbb{O}. Then the claim follows with the computation in Example :

L e 7L e 6L e 5L e 4L e 3L e 2L e 1(1) =e 7(e 6(e 5(e 4(e 3(e 2(e 11)))))) =+1. \begin{aligned} L_{e_7} L_{e_6} L_{e_5} L_{e_4} L_{e_3} L_{e_2} L_{e_1} (1) & = e_7 (e_6 (e_5 (e_4 (e_3 ( e_2 (e_1 1 )))))) \\ & = + 1 \,. \end{aligned}

Basic triples


A special triple or basic triple is a triple (e 1,e 2,e 3)𝕆 3(e_1, e_2, e_3) \in \mathbb{O}^3 of three octonions such that

  • e i 2=1e_i^2 = -1

  • e ie j=e je ie_i e_j = - e_j e_i.

(Whitehead 71, p. 691)


The choice of e 1e_1 identifies an inclusion of the complex numbers 𝕆\mathbb{R} \hookrightarrow \mathbb{C} \hookrightarrow \mathbb{O}.

Then the choice of e 2e_2 on top of that identifies a compatible inclusion of the quaternions 𝕆\mathbb{R} \hookrightarrow \mathbb{C}\hookrightarrow \mathbb{H} \hookrightarrow \mathbb{O}.

Finally the choice of e 3e_3 on top of that induces a basis for all of 𝕆\mathbb{O}.


The set of basic triples, def. , forms a torsor over the automorphism group G2 =Aut(𝕆)= Aut(\mathbb{O}).

(e.g. Baez 02, 4.1)

Relation to quaternions



=1,i,j,k \mathbb{H} = \langle 1, i, j, k\rangle

be the quaternions equipped with canonical basis elements, and let

(2)𝕆= \mathbb{O} = \mathbb{H} \oplus \ell \mathbb{H}

be the octonions equipped with the linear basis induced by the Cayley-Dickson construction (via this def.).

Then the linear map

L L iL jL k:𝕆𝕆 L_{\ell} L_{\ell i} L_{\ell j} L_{\ell k} \;\colon\; \mathbb{O} \longrightarrow \mathbb{O}

is an involution whose +1 eigenspace is \ell \mathbb{H} and whose -1 eigenspace is \mathbb{H}, under the above identification (2).

(Here L ()L_{(-)} denotes the linear map on 𝕆\mathbb{O} given by left multiplication in 𝕆\mathbb{O}.)


We use the Cayley-Dickson relations (this def.)

a(b)=(a¯b),AAa(b)=(ab¯),AA(a)(b 1)=ab¯ a (\ell b) = \ell (\overline{a} b) \,, \phantom{AA} a(\ell b) = (a \overline{b}) \ell \,, \phantom{AA} (\ell a) (b \ell^{-1}) = \overline{a b}

that hold in 𝕆\mathbb{O} for all a,ba,b \in \mathbb{H}, as well as

e=e \ell e = - e \ell

for all imaginary elements ee \in \mathbb{H}.

With this we compute

L L iL jL k(a) =((i)((j)((k)a))) =((i)((j)((k)a))) =((i)((j)((ka¯)))) =+((i)((j)((ka¯) 1))) =+((i)(j(ka¯)¯)) =+((i)(ia¯¯)) =((i)(ia¯¯)) =((iia¯)) =+(a¯) =(a¯ 1) =a \begin{aligned} & L_{\ell} L_{\ell i} L_{\ell j} L_{\ell k} ( a) \\ & = \ell( (\ell i) ( (\ell j) ( (\ell k) a ) ) ) \\ & = - \ell( (\ell i) ( (\ell j) ( (k \ell) a ) ) ) \\ & = - \ell( (\ell i) ( (\ell j) ( (k \overline{a}) \ell ) ) ) \\ & = + \ell( (\ell i) ( (\ell j) ( (k \overline{a}) \ell^{-1} ) ) ) \\ & = + \ell( (\ell i) ( \overline{j (k \overline{a})} ) ) \\ & = + \ell( (\ell i) ( \overline{i \overline{a}} ) ) \\ & = - \ell( (i \ell) ( \overline{i \overline{a}} ) ) \\ & = - \ell( ( i i \overline{a} ) \ell ) \\ & = + \ell( \overline{a} \ell ) \\ & = - \ell( \overline{a} \ell^{-1} ) \\ & = - a \end{aligned}


L L iL jL k(a) =((i)((j)((k)(a)))) =((i)((j)((k)(a¯)))) =((i)((j)(ka¯¯))) =+((i)((j)(ka¯¯))) =+((i)((jka¯))) =((i)((jka¯) 1)) =(ijka¯¯) =+(a¯¯) =+a \begin{aligned} & L_{\ell} L_{\ell i} L_{\ell j} L_{\ell k} ( \ell a) \\ & = \ell( (\ell i) ( (\ell j) ( (\ell k) (\ell a) ) ) ) \\ & = \ell( (\ell i) ( (\ell j) ( (\ell k) ( \overline{a} \ell) ) ) ) \\ & = - \ell( (\ell i) ( (\ell j) ( \overline{k {\overline{a}}} ) ) ) \\ & = + \ell( (\ell i) ( (j \ell) ( \overline{k {\overline{a}}} ) ) ) \\ & = + \ell( (\ell i) ( (j k {\overline{a}}) \ell ) ) \\ & = - \ell( (\ell i) ( (j k {\overline{a}}) \ell^{-1} ) ) \\ & = - \ell( \overline{ i j k {\overline{a}} } ) \\ & = + \ell( \overline{ {\overline{a}} } ) \\ & = + \ell a \end{aligned}

exceptional spinors and real normed division algebras

AA\phantom{AA}spin groupnormed division algebra\,\, brane scan entry
3=2+13 = 2+1Spin(2,1)SL(2,)Spin(2,1) \simeq SL(2,\mathbb{R})A\phantom{A} \mathbb{R} the real numberssuper 1-brane in 3d
4=3+14 = 3+1Spin(3,1)SL(2,)Spin(3,1) \simeq SL(2, \mathbb{C})A\phantom{A} \mathbb{C} the complex numberssuper 2-brane in 4d
6=5+16 = 5+1Spin(5,1)SL(2,)Spin(5,1) \simeq SL(2, \mathbb{H})A\phantom{A} \mathbb{H} the quaternionslittle string
10=9+110 = 9+1Spin(9,1)Spin(9,1) {\simeq}SL(2,O)A\phantom{A} 𝕆\mathbb{O} the octonionsheterotic/type II string


Textbook account:

A survey is in

  • John Baez, The Octonions, Bull. Amer. Math. Soc. 39 (2002), 145-205. (web)

The concept of “special triples” or (“basic triples”) used above seems to go back to

Relation to the Leech lattice:

  • Robert A. Wilson, Octonions and the Leech lattice, Journal of Algebra

    Volume 322, Issue 6, 15 September 2009, Pages 2186-2190, (pdf, slides)

Last revised on August 25, 2019 at 10:23:49. See the history of this page for a list of all contributions to it.