On Real Algebras Admitting Reflections which Commute

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 \(Z_2_2\).