Criteria for Boolean functions to be repetition-free in pre-elementary bases of rank 3

We study representations of Boolean functions by formulas and describe in terms of remainder functions the classes of repetition-free Boolean functions in the pre-elementary bases
\begin{gather*} \{\vee, \cdot, -, 0, 1, x_1x_2x_3 \vee \bar{x}_1\bar{x}_2\bar{x}_3\}, \qquad \{\vee, \cdot, -, 0, 1, x_1(x_2 \vee x_3) \vee x_2x_3\},\\ \{\vee, \cdot, -, 0, 1, x_1(x_2 \vee x_3) \vee x_2\bar{x}_3\}. \end{gather*}