Orbits in the $n$-fold product of a semi-simple complex Jordan algebra

Let $V$ be a complex semi-simple Jordan algebra. Its automorphism group acts on the $n$-fold product of $V$ via the diagonal action. In the talk, this action is studied and a characterization of the closed orbits is given.
In the case of a complex reductive linear algebraic group and the adjoint action on its Lie algebra, the closed orbits are precisely the orbits through semi-simple elements. More generally, a result of R. W. Richardson characterizes the closed orbits of the diagonal action on the $n$-fold product of the Lie algebra. A similar condition can be found in the case of Jordan algebras. It turns out that the orbit through an $n$-tuple $x=(x_1,\ldots, x_n)$ is closed if and only if the Jordan subalgebra generated by $x_1,\ldots, x_n$ is semi-simple.