 Let ${\widehat}B_{0,j}$ denote the [bracket system]{}  obtained from $B_{0,j}$ by replacing every bracket $b \in B_{0,j}$ with ${\widehat}b$. Then ${\widehat}B_{0,0}, \dots, {\widehat}B_{0, \ell_0}$ is a sequence of [bracket system]{}s for $(W, {\Delta})$ that changes the empty [bracket system]{} into $${\widehat}B_{0, \ell_0} = \{  (|p_0|, |p_0|+ |p_0|, 