Editing Alternative algebra (section)

==Occurrence==
The [[projective plane]] over any alternative [[division ring]] is a [[Moufang plane]].

Every [[composition algebra]] is an alternative algebra, as shown by Guy Roos in 2008:<ref>Guy Roos (2008) "Exceptional symmetric domains", §1: Cayley algebras, in ''Symmetries in Complex Analysis'' by Bruce Gilligan & Guy Roos, volume 468 of ''Contemporary Mathematics'', [[American Mathematical Society]]</ref> A composition algebra ''A'' over a field ''K'' has a ''norm n'' that is a multiplicative [[homomorphism]]: <math>n(a \times b) = n(a) \times n(b)</math> connecting (''A'', ×) and (''K'', ×).

Define the form ( _ : _ ): ''A'' × ''A'' → ''K'' by <math>(a:b) = n(a+b) - n(a) - n(b).</math> Then the trace of ''a'' is given by (''a'':1) and the conjugate by ''a''* = (''a'':1)e – ''a'' where e is the basis element for 1. A series of exercises prove that a composition algebra is always an alternative algebra.<ref>{{wikibooks-inline|Associative Composition Algebra/Transcendental paradigm#Categorical treatment}}</ref>