  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.
