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

It is proved that every Rota–Baxter operator of weight zero on the Jordan algebra of a nondegenerate bilinear symmetric form is nilpotent of index less or equal three. We found exact value of nilpotency index of Rota–Baxter operators of weight zero on simple Jordan algebra of Clifford type over the 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.