On Real Algebras Admitting Reflections which Commute

Abstract

We study real algebras admitting reflections which commute. In dimension two, we show that two commuting reflections coincide. We specify the two and four-dimensional real algebras cases. We characterize real algebras of division of two-dimensional to third power associative having a reflection. Finally We give a characterization in four-dimensional, the unitary real algebras of division at third power-associative having two reflections that commute. In eight-dimensional, we give an example of algebra so the group of automorphisms contains a subgroup isomorphic to \(\mathbb{Z}_2\times\mathbb{Z}_2\).