A formula in simple Jordan algebras (Q1065139)

Let A be a (reduced simple) Jordan algebra over a field F of characteristic zero. Let \(e_ 1,...,e_ r\) be a ''primitive decomposition'' of the identity element of A. The main result of this paper is a formula which describes the orbit of \(U=\oplus^{r}_{i=1}F e_ i\) under a maximal connected unipotent subgroup of the structure group of A. Using these results several useful facts about complex simple Jordan algebras and formally real Jordan algebras are derived, several of which have been proven before by somewhat different methods in \textit{H. Braun} and \textit{M. Koecher} [Jordan Algebren (1966; Zbl 0145.260), chapter 11].