Editing Jordan algebra (section)

==Examples==

1. The set of [[self-adjoint]] [[real number|real]], [[complex number|complex]], or [[quaternionic]] matrices with multiplication

:<math>(xy + yx)/2</math>

form a special Jordan algebra.

2. The set of 3×3 self-adjoint matrices over the  [[octonion]]s, again with multiplication

:<math>(xy + yx)/2,</math>

is a 27 dimensional, exceptional Jordan algebra (it is exceptional because the [[octonion]]s are not associative). This was the first example of an [[Albert algebra]]. Its automorphism group is the exceptional [[Lie group]] [[F4 (mathematics)|F<sub>4</sub>]]. Since over the [[complex numbers]] this is the only simple exceptional Jordan algebra up to isomorphism,<ref name=Springer00/> it is often referred to as "the" exceptional Jordan algebra. Over the [[real numbers]] there are three isomorphism classes of simple exceptional Jordan algebras.<ref name=Springer00>{{harvnb|Springer|Veldkamp|2000|loc=§5.8, p.&nbsp;153}}</ref>