Clifford translations of symmetric spaces

Abstract:

A direct proof, not using the classification of symmetric spaces, is given for the following characterization of Clifford translations in a symmetric space $M$: An isometry $g$ is a Clifford translation of $M$ if and only if the centralizer $Z(g)$ of $g$ in the isometry group of $M$ is transitive on $M$. The proof uses a geodesic characterization of Clifford translations, and the subgroups ${T^{(h)}}$ of J. de Siebenthal.