Isotopes of Octonion Algebras, G2-Torsors and Triality

Abstract: Octonion algebras over rings are, in contrast to those over fields, not determined by their norm forms. Octonion algebras whose norm is isometric to the norm q of a given algebra C are twisted forms of C by means of the Aut(C)-torsor O(q) ->O(q)/Aut(C). We show that, over any commutative unital ring, these twisted forms are precisely the isotopes C(a,b) of C, with multiplication given by x*y=(xa)(by), for unit norm octonions a,b of C. The link is provided by the triality phenomenon, which we study from new and classical perspectives. We then study these twisted forms using the interplay, thus obtained, between torsor geometry and isotope computations, thus obtaining new results on octonion algebras over e.g. rings of (Laurent) polynomials.