Proposition 8

Let $(S_i)_{i \in I}$ be a partition of $S$ such that every element of $S_i$ commutes with every element of $S_j$ if $i \neq j$. For all $i \in I$, let $W_i$ be the subgroup generated by $S_i$. Then $W$ is the restricted direct product of the family $(W_i)_{i \in I}$.

It is clear that for all $i \in I$ the subgroup $W'_i$ generated by the union of the $W_j$ for $j \neq i$ is also generated by $S'_i = \bigcup_{j \neq i} S_j$. Thus

by Th. 2 of no. 8. Since $W$ is generated by the union of the $W_i$, this proves the proposition.