Canonical form of element of Jordan algebra by group \(\text{Spin}(9)\) (Q2906179)

Consider the real exceptional Jordan algebra \(J=H_3(O,*)\) where \(O\) denotes the real octonion division algebra and \(*\) is the standard involution. It is known that any element in \(J\) can be transformed to a diagonal form by the natural action of some element in the Lie group \(F_4\). In the paper under review it is found a canonical form for elements in \(J\) under the action of \(\text{Spin}(9)\) which is a maximal subgroup of the compact exceptional Lie group \(F_4\) of maximal rank. The paper also contains a result on the canonical form of elements in another exceptional Jordan algebra whose group of automorphisms is the connected noncompact exceptional Lie group \(F_{4(-20)}\) which contains also a copy of \(\text{Spin}(9)\).