1,662
edits
(reduce verbosity per Wikipedia:Manual_of_Style/Words_to_watch#Editorializing and MOS:NOTE) 

===Definition===
A '''Hurwitz algebra''' or '''composition algebra''' is a finitedimensional not necessarily associative algebra {{mvarA}} with identity endowed with a nondegenerate quadratic form {{mvarq}} such that {{math1=''q''(''a b'') = ''q''(''a'') ''q''(''b'')}}. If the underlying coefficient field is the [[real numberreals]] and {{mvarq}} is positivedefinite, so that {{math1=(''a'', ''b'') = {{sfrac12}}[''q''(''a'' + ''b'') − ''q''(''a'') − ''q''(''b'')]}} is an [[inner product spaceinner product]], then {{mvarA}} is called a '''Euclidean Hurwitz algebra''' or (finitedimensional) '''normed division algebra'''.<ref>{{harvnbFarautKoranyi1994p=82}}</ref>
If {{mvarA}} is a Euclidean Hurwitz algebra and {{mvara}} is in {{mvarA}}, define the involution and right and left multiplication operators by
