The category of all 2-dimensional real division algebras is shown to split into four full subcategories, each of which is given by the natural action of a Coxeter group of type 𝔸1 or 𝔸2 on the set of all pairs of ellipses in ℝ2, which are centred in the origin and have reciprocal axis lengths. Cross-sections for the orbit sets of these group actions are being determined. They yield a classification of all 2-dimensional real division algebras. Moreover all morphisms between the objects in this classifying list are described, and thus an explicit and geometric picture of the category of all 2-dimensional real division algebras is obtained. This elementary and self-contained exposition extends Darpö and Dieterich's recent description [14] of the category of all 2-dimensional commutative real division algebras, which in turn is based on Benkart, Britten and Osborn's investigation [4] of the isotopes of ℂ. It also supplements earlier contributions of Althoen and Kugler [2], Burdujan [9], Gottschling [25], Petersson [33], Hübner and Petersson [29], and Doković and Zhao [23] to the problem of classifying all 2-dimensional real division algebras.
The Automorphism Groups and Derivation Algebras of Two-Dimensional Algebras
