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Let $J$ be a finite-dimensional semi-simple Jordan algebra over analgebraically closed field of characteristic $0$. In this article, the diagonalaction 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 subalgebragenerated by the elements $x_1,\ldots, x_n$ is semi-simple.

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