Editing Octonion (section)

==Arithmetic and operations==
===Addition and subtraction===
Addition and subtraction of octonions is done by adding and subtracting corresponding terms and hence their coefficients, like quaternions.

===Multiplication===
Multiplication of octonions is more complex. Multiplication is [[Distributive property|distributive]] over addition, so the product of two octonions can be calculated by summing the products of all the terms, again like quaternions. The product of each pair of terms can be given by multiplication of the coefficients and a [[multiplication table]] of the unit octonions, like this one (given both by [[Arthur Cayley]] in 1845 and [[John T. Graves]] in 1843):<ref name=GSSV>
{{cite book
 |first1=G.   |last1=Gentili          |first2=C.  |last2=Stoppato
 |first3=D.C. |last3=Struppa          |first4=F.  |last4=Vlacci
 |year=2009
 |chapter=Recent developments for regular functions of a hypercomplex variable
 |editor1-first=I. |editor1-last=Sabadini |editor1-link=Irene Sabadini
 |editor2-first=M. |editor2-last=Shapiro
 |editor3-first=F. |editor3-last=Sommen
 |title=Hypercomplex Analysis
 |publisher=[[Birkhäuser]]
 |isbn=978-3-7643-9892-7
 |page=168
 |chapter-url=https://books.google.com/books?id=H-5v6pPpyb4C&pg=PA168 |via=Google books
}}
</ref>

{|class="wikitable" style="text-align: center; margin:0.5em auto"
|-
  !colspan="2" rowspan="2"| <math>e_ie_j</math>
  !colspan="8" |<math>e_j</math>
|-
  ! width="30pt" | <math>e_0</math>
  ! width="30pt" | <math>e_1</math>
  ! width="30pt" | <math>e_2</math>
  ! width="30pt" | <math>e_3</math>
  ! width="30pt" | <math>e_4</math>
  ! width="30pt" | <math>e_5</math>
  ! width="30pt" | <math>e_6</math>
  ! width="30pt" | <math>e_7</math>
|-
  !rowspan="8" |<math>e_i</math>
  !<math>e_0</math>
  |<math>e_0</math>
  |<math>e_1</math>
  |<math>e_2</math>
  |<math>e_3</math>
  |<math>e_4</math>
  |<math>e_5</math>
  |<math>e_6</math>
  |<math>e_7</math>
|-
  !<math>e_1</math>
  |<math>e_1</math>
  |<math>-e_0</math>
  |<math>e_3</math>
  |<math>-e_2</math>
  |<math>e_5</math>
  |<math>-e_4</math>
  |<math>-e_7</math>
  |<math>e_6</math>
|-
  !<math>e_2</math>
  |<math>e_2</math>
  |<math>-e_3</math>
  |<math>-e_0</math>
  |<math>e_1</math>
  |<math>e_6</math>
  |<math>e_7</math>
  |<math>-e_4</math>
  |<math>-e_5</math>
|-
  !<math>e_3</math>
  |<math>e_3</math>
  |<math>e_2</math>
  |<math>-e_1</math>
  |<math>-e_0</math>
  |<math>e_7</math>
  |<math>-e_6</math>
  |<math>e_5</math>
  |<math>-e_4</math>
|-
  !<math>e_4</math>
  |<math>e_4</math>
  |<math>-e_5</math>
  |<math>-e_6</math>
  |<math>-e_7</math>
  |<math>-e_0</math>
  |<math>e_1</math>
  |<math>e_2</math>
  |<math>e_3</math>
|-
  !<math>e_5</math>
  |<math>e_5</math>
  |<math>e_4</math>
  |<math>-e_7</math>
  |<math>e_6</math>
  |<math>-e_1</math>
  |<math>-e_0</math>
  |<math>-e_3</math>
  |<math>e_2</math>
|-
  !<math>e_6</math>
  |<math>e_6</math>
  |<math>e_7</math>
  |<math>e_4</math>
  |<math>-e_5</math>
  |<math>-e_2</math>
  |<math>e_3</math>
  |<math>-e_0</math>
  |<math>-e_1</math>
|-
  !<math>e_7</math>
  |<math>e_7</math>
  |<math>-e_6</math>
  |<math>e_5</math>
  |<math>e_4</math>
  |<math>-e_3</math>
  |<math>-e_2</math>
  |<math>e_1</math>
  |<math>-e_0</math>
|}

Most off-diagonal elements of the table are antisymmetric, making it almost a [[skew-symmetric matrix]] except for the elements on the main diagonal, as well as the row and column for which {{math|''e''<sub>0</sub>}} is an operand.

The table can be summarized as follows:<ref name= Shestakov>

{{cite book |first1=L.V. |last1=Sabinin |first2=L. |last2=Sbitneva |first3=I.P. |last3=Shestakov |year=2006 |chapter=§17.2 Octonion algebra and its regular bimodule representation |title=Non-Associative Algebra and its Applications |place=Boca Raton, FL |publisher=CRC Press |isbn=0-8247-2669-3 |page=235 |chapter-url=https://books.google.com/books?id=_PEWt18egGgC&pg=PA235 |via=Google books }}</ref>

: <math>
e_\ell e_m  =
\begin{cases}
e_m , & \text{if }\ell = 0 \\
e_\ell , & \text{if }m = 0 \\
- \delta_{\ell m}e_0 + \varepsilon _{\ell m n} e_n, & \text{otherwise}
\end{cases}
</math>

where {{mvar|δ<sub>ℓm</sub>}} is the [[Kronecker delta]] (equal to {{math|1}} if {{math|''ℓ'' {{=}} ''m''}}, and {{math|0}} for {{math|''ℓ'' ≠ ''m''}}), and {{mvar|ε<sub>ℓmn</sub>}} is a [[completely antisymmetric tensor]] with value {{math|+1}} when {{math| {{nobr| ''ℓ m n''}} {{=}} {{nobr| 1 2 3,}} {{nobr| 1 4 5,}} {{nobr| 1 7 6,}} {{nobr| 2 4 6,}} {{nobr| 2 5 7,}}  {{nobr| 3 4 7,}}  {{nobr| 3 6 5 ,}} }} and any even number of [[permutation]]s of the indices, but {{math|−1}} for any odd [[permutation]]s of the listed triples (e.g. <math>\ \varepsilon_{1 2 3} = +1\ </math> but <math>\ \varepsilon_{1 3 2} = \varepsilon_{2 1 3} = -1\ ,</math> however, <math>\ \varepsilon_{3 1 2} = \varepsilon_{2 3 1} = +1\ </math> again). Whenever any two of the three indices are the same, {{nobr| {{mvar|ε<sub>ℓmn</sub>}} {{math|{{=}} 0}} .}}

The above definition is not unique, however; it is only one of 480&nbsp;possible definitions for octonion multiplication with {{math|''e''<sub>0</sub> {{=}} 1}}. The others can be obtained by permuting and changing the signs of the non-scalar basis elements {{math|{{big|{}}''e''<sub>1</sub>, ''e''<sub>2</sub>, ''e''<sub>3</sub>, ''e''<sub>4</sub>, ''e''<sub>5</sub>, ''e''<sub>6</sub>, ''e''<sub>7</sub>{{big|}<nowiki/>}}&nbsp;.}} The 480&nbsp;different algebras are [[isomorphism|isomorphic]], and there is rarely a need to consider which particular multiplication rule is used.

Each of these 480&nbsp;definitions is invariant up to signs under some 7&nbsp;cycle of the points {{nobr|{{math| (1 2 3 4 5 6 7)}} ,}} and for each 7&nbsp;cycle there are four definitions, differing by signs and reversal of order. A common choice is to use the definition invariant under the 7&nbsp;cycle (1234567) with {{math|''e''<sub>1</sub>''e''<sub>2</sub> {{=}} ''e''<sub>4</sub>}} by using the triangular multiplication diagram, or Fano plane below that also shows the sorted list of {{nobr|1 2 4}} based 7-cycle triads and its associated multiplication matrices in both {{math|''e''<sub>''n''</sub>}} and <math>\ \mathrm{IJKL}\ </math> format. 
:[[File:FanoPlane_with_GeometricAlgebra.svg|900px|Octonion triads, Fano plane, and multiplication matrices]]

A variant of this sometimes used is to label the elements of the basis by the elements {{math|∞}}, 0, 1, 2, ..., 6, of the [[projective line]] over the [[finite field]] of order 7. The multiplication is then given by {{math|''e''<sub>∞</sub> {{=}} 1}} and {{math|''e''<sub>0</sub>''e''<sub>1</sub> {{=}} ''e''<sub>3</sub>}}, and all equations obtained from this one by adding a constant ([[modular arithmetic|modulo]] 7) to all subscripts: In other words using the seven triples {{nobr|(0 1 3), {{nobr|(1 2 4)}}, {{nobr|(2 3 5)}}, {{nobr|(3 4 6)}}, {{nobr|(4 5 0)}}, {{nobr|( 5 6 1)}}, {{nobr|(6 0 2)}}  .}} These are the nonzero codewords of the [[quadratic residue code]] of length&nbsp;7 over the [[Finite field|Galois field]] of two elements, {{math|[[GF(2)|''GF''(2)]]}}. There is a symmetry of order&nbsp;7 given by adding a constant [[modulo arithmetic|mod]]&nbsp;7 to all subscripts, and also a symmetry of order 3 given by multiplying all subscripts by one of the quadratic residues 1, 2, 4 mod&nbsp;7&nbsp;.<ref name=Parra>
{{cite book
 |first1=Rafał    |last1=Abłamowicz
 |first2=Pertti   |last2=Lounesto
 |first3=Josep M. |last3=Parra
 |year=1996
 |chapter=§&nbsp;Four ocotonionic basis numberings
 |title=Clifford Algebras with Numeric and Symbolic Computations
 |publisher=Birkhäuser
 |isbn=0-8176-3907-1
 |page=202
 |chapter-url=https://books.google.com/books?id=OpbY_abijtwC&pg=PA202 |via=Google books
}}
</ref><ref name=Manogue>
{{cite journal
 |first1=Jörg       |last1=Schray
 |first2=Corinne A. |last2=Manogue
 |date=January 1996
 |title=Octonionic representations of Clifford algebras and triality
 |journal=Foundations of Physics
 |volume=26 |issue=1 |pages=17–70
 |doi=10.1007/BF02058887 |arxiv=hep-th/9407179
 |bibcode=1996FoPh...26...17S |s2cid=119604596
}}
: Available as {{cite journal |title=Octonionic representations of Clifford algebras and triality |date=1996 |doi=10.1007/BF02058887 |arxiv=hep-th/9407179 |last1=Schray |first1=Jörg |last2=Manogue |first2=Corinne A. |journal=Foundations of Physics |volume=26 |issue=1 |pages=17–70 |bibcode=1996FoPh...26...17S }}, in particular {{cite AV media |title=Figure&nbsp;1 |medium=image |format=[[.png]] |website=[[arXiv]] |url=https://arxiv.org/PS_cache/hep-th/ps/9407/9407179v1.fig1-1.png }}
</ref>
These seven triples can also be considered as the seven translates of the set {1,2,4} of non-zero squares 
forming a cyclic (7,3,1)-[[difference set]] in the finite field {{math|[[GF(7)]]}} of seven elements. 

The Fano plane shown above with <math>e_n</math> and IJKL multiplication matrices also includes the [[geometric algebra]] basis with signature {{nobr|{{math|(− − − −)}}}} and is given in terms of the following 7&nbsp;[[quaternion]]ic triples (omitting the scalar identity element):
:{{math|(''I'' , ''j'' , ''k'' ) , ( ''i'' , ''J'' , ''k'') , ( ''i'' , ''j'' , ''K'') , (''I'' , ''J'' , ''K'' ) , ([[Hodge star operator|★]]''I'' , ''i'' , ''l'' ) , (★''J'' , ''j'' , ''l'' ), (★''K'' , ''k'' , ''l'')}}

or alternatively:

:{{math|<math>(\sigma_{1},j,k),(i,\sigma_{2},k),(i,j,\sigma_{3}),(\sigma_{1},\sigma_{2},\sigma_{3}),</math>([[Hodge star operator|★]]<math>\sigma_{1},i,l),(</math>★<math>\sigma_{2},j,l),(</math>★<math>\sigma_{3},k,l)</math>}}

in which the lower case items ''{i, j, k, l}'' are [[vector (mathematics and physics)|vectors]] (e.g. {<math>\gamma_{0},\gamma_{1},\gamma_{2},\gamma_{3}</math>}, respectively) and the upper case ones {''I,J,K''}={''σ<sub>1</sub>,σ<sub>2</sub>,σ<sub>3</sub>''} are [[bivector]]s (e.g. <math>\gamma_{\{1,2,3\}}\gamma_{0}</math>, respectively)  and the [[Hodge star operator]] {{math|[[Hodge star operator|★]] {{=}} ''i j k l''}} is the pseudo-scalar element. If the {{math|★}} is forced to be equal to the identity, then the multiplication ceases to be associative, but the {{math|★}} may be removed from the multiplication table resulting in an octonion multiplication table.

In keeping {{math|[[Hodge star operator|★]] {{=}} ''i j k l''}} associative and thus not reducing the 4&nbsp;dimensional geometric algebra to an octonion one, the whole multiplication table can be derived from the equation for {{math|★}}. Consider the [[gamma matrices]] in the examples given above. The formula defining the fifth gamma matrix (<math>\gamma_{5}</math>) shows that it is the {{math|★}} of a four-dimensional geometric algebra of the gamma matrices.

===Fano plane mnemonic===
[[File:FanoPlane.svg|thumb|A mnemonic for the products of the unit octonions<ref name="Baez 2002 loc=p. 6">{{Harv|Baez|2002|loc=p. 6}}</ref>]]
[[File:Octonion-Fano Cube.gif|thumb|A 3D mnemonic visualization showing the 7 triads as [[hyperplane]]s through the real ({{math|''e''<sub>0</sub>}}) vertex of the octonion example given above<ref name="Baez 2002 loc=p. 6"/>]]
A convenient [[mnemonic]] for remembering the products of unit octonions is given by the diagram, which represents the multiplication table of Cayley and Graves.<ref name=GSSV/><ref name=Ablamowicz>
{{cite book
 |first1=Tevian     |last1=Dray
 |first2=Corinne A. |last2=Manogue
 |name-list-style=amp
 |year=2004
 |chapter=Chapter&nbsp;29: Using octonions to describe fundamental particles
 |title=Clifford Algebras: Applications to mathematics, physics, and engineering
 |editor1-first=Rafał |editor1-last=Abłamowicz
 |publisher=[[Birkhäuser]]
 |isbn=0-8176-3525-4
 |at=Figure&nbsp;29.1: Representation of multiplication table on projective plane. p.&nbsp;452
 |chapter-url=https://books.google.com/books?id=b6mbSCv_MHMC&pg=PA452 |via=Google books
}}
</ref>
This diagram with seven points and seven lines (the circle through 1, 2, and 3 is considered a line) is called the [[Fano plane]]. The lines are directional. The seven points correspond to the seven standard basis elements of <math>\ \operatorname\mathcal{I_m}\bigl[\ \mathbb{O}\ \bigr]\ </math> (see definition [[#Conjugate, norm, and inverse|below]]). Each pair of distinct points lies on a unique line and each line runs through exactly three points.

Let {{math|(''a'', ''b'', ''c'')}} be an ordered triple of points lying on a given line with the order specified by the direction of the arrow. Then multiplication is given by

:{{math|''ab'' {{=}} ''c''}} and {{math|''ba'' {{=}} −''c''}}

together with [[cyclic permutation]]s. These rules together with

* {{math|1}} is the multiplicative identity,
* <math>{e_i}^2 = -1\ </math> for each point in the diagram

completely defines the multiplicative structure of the octonions. Each of the seven lines generates a [[Subalgebra#Subalgebras for algebras over a ring or field|subalgebra]] of <math>\ \mathbb{O}\ </math> isomorphic to the quaternions {{math|'''H'''}}.

===Conjugate, norm, and inverse===
The ''conjugate'' of an octonion

:<math> x = x_0\ e_0 + x_1\ e_1 + x_2\ e_2 + x_3\ e_3 + x_4\ e_4 + x_5\ e_5 + x_6\ e_6 + x_7\ e_7 </math>

is given by

:<math> x^* = x_0\ e_0 - x_1\ e_1 - x_2\ e_2 - x_3\ e_3 - x_4\ e_4 - x_5\ e_5 - x_6\ e_6 - x_7\ e_7 ~.</math>
Conjugation is an [[involution (mathematics)|involution]] of <math>\ \mathbb{O}\ </math> and satisfies {{math|(''xy'')* {{=}} ''y''*''x''*}} (note the change in order).

The ''real part'' of {{mvar|x}} is given by

:<math>\frac{x + x^*}{2} = x_0\ e_0</math>

and the ''imaginary part'' (sometimes called the ''pure part'') by

:<math> \frac{x - x^*}{2} = x_1\ e_1 + x_2\ e_2 + x_3\ e_3 + x_4\ e_4 + x_5\ e_5 + x_6\ e_6 + x_7\ e_7 ~.</math>

The set of all purely imaginary octonions [[linear span|spans]] a 7&nbsp;[[dimension (vector space)|dimensional]] [[linear subspace|subspace]] of <math>\ \mathbb{O}\ ,</math> denoted <math>\ \operatorname\mathcal{I_m}\bigl[\ \mathbb{O}\ \bigr] ~.</math>

Conjugation of octonions satisfies the equation

:<math> -6 x^* = x + (e_1x)e_1 + (e_2x)e_2 + (e_3x)e_3 + (e_4x)e_4 + (e_5x)e_5 + (e_6x)e_6 + (e_7x)e_7 ~.</math>

The product of an octonion with its conjugate, {{nobr| {{math|''x''*''x'' {{=}} ''xx''*}} ,}} is always a nonnegative real number:

:<math>x^*x = x_0^2 + x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 + x_7^2 ~.</math>

Using this, the norm of an octonion is defined as

:<math>\|x\| = \sqrt{x^*x} ~.</math>

This norm agrees with the standard 8&nbsp;dimensional [[Euclidean norm]] on {{math|ℝ<sup>8</sup>}}.

The existence of a norm on <math>\ \mathbb{O}\ </math> implies the existence of [[inverse element|inverses]] for every nonzero element of <math>\ \mathbb{O} ~.</math> The inverse of{{nobr| {{math| ''x'' ≠ 0}} ,}} which is the unique octonion {{math|''x''<sup>−1</sup>}} satisfying {{nobr|{{math| ''x x''<sup>−1</sup> {{=}} ''x''<sup>−1</sup>''x'' {{=}} 1}} ,}} is given by

:<math>x^{-1} = \frac {x^*}{\|x\|^2} ~.</math>

===Exponentiation and polar form===
Any octonion {{mvar|x}} can be decomposed into its real part and imaginary part:

<math>x=\mathfrak{R}(x)+\mathfrak{I}(x)</math>

also sometimes called scalar and vector parts.

We define the ''unit vector'' {{mvar|u}} corresponding to {{mvar|x}} as

<math>u=\frac{\mathfrak{I}(x)}{\|\mathfrak{I}(x)\|}</math>. It is a pure octonion of norm 1.

It can be proved<ref> {{cite web|url=https://mathsci.kaist.ac.kr/~tambour/fichiers/publications/Ensembles_de_nombres.pdf|date=6 September 2011|title=Ensembles de nombres|publisher=Forum Futura-Science|access-date=24 February 2025|language=fr}}</ref> that any non-zero octonion can be written as:

<math>o=\|o\|(\cos\theta+u\sin\theta)=\|o\|e^{u\theta}</math>

thus providing a polar form.