  Let $J$ be a finite-dimensional semi-simple Jordan algebra over an
algebraically closed field of characteristic $0$. In this article, the diagonal
action of the automorphism group of $J$ on the $n$-fold product $J\times\ldots
\times J$ is studied. In particular, it is shown that the orbit through an
$n$-tuple $x=(x_1,,\ldots,x_n)$ is closed if and only if the Jordan subalgebra
generated by the elements $x_1,\ldots, x_n$ is semi-simple.
