Editing Octonion (section)

===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'''}}.