In the article, we give an algorithm for constructing the Pauli operator. It is shown that the Pauli construction is connected with the problem of zero divisors in Clifford algebras. Considered cases when a new basis is formed by elements of the first rank. Let $R_{p,q}$ be the real Clifford algebra of dimension $m = 2^{n}$, $n=p+q$. Let $e_0=1$ be the identity element and let $e_1,e_2,...,e_n$ be generating basis of Clifford algebra: \[e_ie_j+e_je_i=2{\delta }_{ij}{\varepsilon }_i,\] where ${\varepsilon }_i=e^{2}_i=1, i=1,...,p,{\varepsilon }_i=e^{2}_i=-1,\ \ i=p+1,...,p+q$. Elements $e_{\alpha } =e_{i_{1} }...e_{i_{k} }, 1\le i_{1} <...< i_{k} \le n$ together with the identity element $e_0$ form the basis of Clifford algebra. An arbitrary element of $w\in R_{p,q}$ can be written as the sum of the ranks of elements of 0 to $n$: $w=\sum^n_{k=0}{}w^{k}.$ Three involution operations are known: a reversal involution, a grade involution, a Clifford involution. The Clifford involution is a superposition of the reversal and the grade involution: $w\to \overline{w}=\widehat{\tilde{w}}.$ For the element $w\in R_{p,q}$ we have: $\overline{w}=\sum^n_{k=0}{}(-1{)}^{\frac{k(k+1)}{2}}w^{k}.$ Consider another generating basis $\{ e_{\gamma _{i} } \} _{i=1}^{n} ,$ formed by the elements $e_{\gamma _{i} } =\sum _{k=1}^{n}\alpha _{k}^{(i)} e_{k},$ for which the above conditions are fulfilled. Let us find Pauli's operator $T=\sum_{\alpha \in {\Gamma }_n}{}a_{\alpha }e_{\alpha}$$ $, which has an inverse $T^{-1}$ and satisfies the equations: \[Te_i=e_{{\gamma }_i}T,i=1,2,...,n.\] The construction of the operator $T$ consists of $n$ steps, Clifford pairing is applied here. Theorem 1. Let in even Clifford algebra $R_{p,q}$, elements of a new basis be represented as $e_{\gamma _{i} } =\sum _{k=1}^{n}\alpha _{k}^{(i)} e_{k} ,i=1,2,...,n. $ Then there exists an unique (up to multiplication by a real number) element of a Clifford algebra $T$, such that $e_{{\gamma }_i}=Te_iT^{-1},i=1,2,\dots ,n.$ Operator Pauli is an iterative formula $T=T_n,$ where \[T_{1}=e_0\pm {\varepsilon }_{1}e_{{\gamma }_{1}}e_{1},\] plus sign is taken if ${\alpha }^{(1)}_{1}\neq -1,$ minus sign if ${\alpha }^{(1)}_{1}=-1,$ \[T_i=T_{i-1}\pm {\varepsilon }_ie_{{\gamma }_i}T_{i-1}e_i,i=2,3,\dots ,n,\] plus sign is taken if $({\overline{T}}_{i-1}T_i+{\overline{T}}_iT_{i-1})\neq 0,$ minus sign if $({\overline{T}}_{i-1}T_i+{\overline{T}}_iT_{i-1})=0$, ${\varepsilon }_i=e^{2}_i=e^{2}_{{\gamma }_i}.$ This algorithm is used for construction of the Pauli operator in an odd Clifford algebra.