Editing Octonion (section)

===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.