  This article discusses the representation theory of noncommutative algebras
reality-based algebras with positive degree map over their field of definition.
When the standard basis contains exactly two nonreal elements, the main result
expresses the noncommutative simple component as a generalized quaternion
algebra over its field of definition. The field of real numbers will always be
a splitting field for this algebra, but there are noncommutative table algebras
of dimension $6$ with rational field of definition for which it is a division
algebra. The approach has other applications, one of which shows noncommutative
association scheme of rank $7$ must have at least three symmetric relations.
