Conjugacy classes of trialitarian automorphisms and symmetric compositions. (Q2839721)

Let \((S,n)\) be an eight-dimensional vector space over an arbitrary field \(F\), endowed with a \(3\)-fold Pfister form \(n\). The main result of the paper under review sets up carefully a one-to-one correspondence between conjugacy classes of trialitarian automorphisms of \(\mathbf{PGO}^+(n)\) (i.e., outer automorphisms of order \(3\) defined over \(F\)) and isomorphism classes of symmetric compositions \(*\) defined on \((S,n)\) (i.e.; bilinear multiplications satisfying \(n(x*y)=n(x)n(y)\) and \(n(x*y,z)=n(x,y*z)\) for any \(x,y,z\in S\), where \(n(x,y)\) is the polar form of \(n\)). An analogous result is proved, too, for the simply connected group \(\mathbf{Spin}(n)\).NEWLINENEWLINE Given any symmetric composition algebra \((S,*,n)\) and any similarity \(f\in\text{GO}^+(n)\), there are similarities \(g,h\in\text{GO}^+(n)\) and a nonzero scalar \(\lambda\) such that \(\lambda f(x*y)=g(x)*h(y)\) for any \(x,y\in S\). The assignment \(\rho_*\colon[f]\mapsto [g]\) gives a trialitarian automorphism of \(\mathbf{PGO}^+(n)\). This is how the bijection above is constructed.NEWLINENEWLINE As a consequence, there are two types of conjugacy classes of trialitarian automorphisms. In the first type, the fixed subgroup is connected and simple of type \(G_2\). In the second type, this subgroup is adjoint of type \(A_2\) if the characteristic is not \(3\), while it is not smooth in characteristic \(3\).