Conjugacy classes of $n$-tuples in semi-simple Jordan algebras

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.