Pauli's theorem in Clifford algebras of odd dimension

Pauli's theorem is investigated in real Clifford algebras of odd dimension. In Clifford algebras $R_{3,0}$ and $R_{5,0}$ an algorithm for constructing the Pauli operator is given. This algorithm is transferred to an arbitrary Clifford algebra of odd dimension $R_{p,q+1}$ ($R_{p+1,q}$). An iterative formula for finding the Pauli operator is obtained. It is shown that the problem of constructing the Pauli operator is related to the problem of zero divisors in Clifford algebras. When constructing Pauli operators, two types of conjugations are used: Clifford conjugation and reverse conjugation. If $p+q+1\equiv 3 \pmod 4$, then when constructing the Pauli operator Clifford conjugation is used; if $p+q+1\equiv 1 \pmod 4$ then reverse conjugation is used.