Isotopes of octonion algebras, \(\mathbf{G}_{2}\)-torsors and triality (Q1711931)

Given a field \(k\), or more generally a local ring, two octonion algebras over \(k\) are isomorphic if and only if their quadratic forms are isometric. This was proved to be false over more general commutative rings by \textit{P. Gille} [Can. Math. Bull. 57, No. 2, 303--309 (2014; Zbl 1358.11126)]. Let \(C\) be an octonion algebra over a unital commutative ring \(R\), with norm \(q\). The paper under review deals with those octonion algebras over \(R\) whose norm is isometric to \(q\). The main result shows that, up to isomorphism, these octonion algebras are the isotopes \(C^{a,b}\) for norm \(1\) elements \(a,b\in C\), which are defined on \(C\), with the same norm, but with new multiplication \(x*y=(xa)(by)\). The key ingredient to achieving this is triality. The group scheme \(\mathbf{RT}(C)\), whose \(S\)-points are the triples \((t_1,t_2,t_3)\in \mathbf{SO}(q)(S)\) such that \(t_1(xy)=\overline{t_2(\bar x)}\,\overline{t_3(\bar y)}\) for any \(S\)-ring \(T\) and any \(x,y\in C_T\), has cyclic symmetry, and it is isomorphic to \(\mathbf{Spin}(q)\). The group scheme \(\mathbf{RT}(C)\) acts naturally on two copies of the unit sphere: \(\mathbf{S}_C^2\), and the stabilizer of \((1,1)\) is isomorphic to the automorphism group scheme \(\mathbf{Aut}(C)\). Moreover, the fppf quotient sheaf \(\mathbf{RT}(C)/\mathbf{Aut}(C)\) turns out to be isomorphic to \(\mathbf{S}_C^2\). This gives a very precise description of the \(\mathbf{Aut}(C)\)-torsor \(\Pi:\mathbf{Spin}(q)\rightarrow\mathbf{Spin}(q)/\mathbf{Aut}(C)\). It is shown that the \(\Pi\)-twists of \(C\) correspond canonically to the isotopes \(C^{a,b}\) above, and the main result follows by proving that the torsor \(\Pi\) gives the same objects as \(\mathbf{O}(q)\rightarrow \mathbf{O}(q)/\mathbf{Aut}(C)\). From this general framework, new results are deduced for some particular rings, like the rings of (Laurent) polynomials.