Rota-Baxter operators of zero weight on simple Jordan algebra of Clifford type

It is proved that any Rota—Baxter operator of zero weight on Jordan algebra of a nondegenerate bilinear symmetric form is nilpotent of index less or equal three. We state exact value of nilpotency index on simple Jordan algebra of Clifford type over fields $\mathbb{R}$, $\mathbb{C}$, and ${\mathbb{Z}}_p$. For ${\mathbb{Z}}_p$, we essentially use the results from number theory concerned quadratic residues and Chevalley—Warning theorem.