Orbit decomposition of Jordan matrix algebras of order three under the automorphism groups (Q2902445)

Let \(\mathcal{J}'\) be a split exceptional simple Jordan algebra over a field \(\mathbb{F}\) of characteristic not two, that is, the set of all hermitian matrices of order three whose elements are split octonions over \(\mathbb{F}\) with the Jordan product. And let \(G'\) be the automorphism group of \(\mathcal{J}'\).NEWLINENEWLINEIn this paper the authors present a concrete orbit decomposition under the automorphism group on a real split Jordan algebra of all hermitian matrices of order three corresponding to any real split composition algebra, or the complexification of it, that is special or exceptional as a Jordan algebra. This implies that \(X,Y \in \mathcal{J}'\) are in the same \(G'\)-orbit if and only if \(X,Y\) admit the same dimension of the generating subspace with \(E\) by the cross product and the same characteristic polynomial, which gives a simplification for N. Jacobson's polynomial invariants on \(G'\)-orbits when \(\mathbb{F}=\mathbb{R}\) or the field of all complex number \(\mathbb{C}\).