{{R}}_{{{\alpha}},4k+2} & \hbox{ if ${{\alpha}}$ is a short root};\\
     {{R}}_{{{\alpha}},2k}& \hbox{ if ${{\alpha}}$ is a long root};\\ 
     {{R}}_{{{\alpha}},2t_{{\alpha}}k+1} & \hbox{ if ${{\alpha}}>r$};\end{array}\right.$$

2.  The mutation matrix $B^{(1)}$ at $k^{(1)}$ has the following form: $$B^{(1)} = \left( \begin{array}{c|c} -A & -D^t \\ \hline D &
        2A\end{array}\right),$$ where $$D_{ij} = \left\{ \begin{array}{ll}
     C_{ij}, & \hbox{ if $i$ and $j$ are both long roots}; \\
    -C_{ij}, & \hbox{ if $j$ is short}; \\
    0 & \hbox{otherwise.}\end{array}\right.$$

The first statement follows from the discussion above.

The second statement involves several cases.

-   Because the mutations $\mu_{{\alpha}}$ with ${{\alpha}}<r$ and ${{\alpha}}\in \Pi_<$ mutually commute, their effect on the columns and rows of $B$ indexed by $\{{{\alpha}}:~{{\alpha}}\in
    \Pi_<\}$ is only to change their sign. Thus, $$B_{i,{{\alpha}}}^{(1)}=-B_{{{\alpha}},i}^{(1)}=-B_{i,{{\alpha}}},\quad {{\alpha}}\in \Pi_<.$$ This accounts for the appearance of $-A$ in the upper left hand corner of $B^{(1)}$, as well as the columns of the matrix $D$ which correspond to short roots.

-   Let $\beta\in \Pi_>$ be a long root and consider the change column $\beta$ of $B$. If $i\notin \Pi_<$, then $$B_{i,\beta}^{(1)} = B_{i,\beta} + \sum_{{{\alpha}}\in \Pi_<}
    {{\rm sgn}}(B_{i,{{\alpha}}}) [B_{i,{{\alpha}}}B_{{{\alpha}},\beta}]_+.$$ But $B_{{{\alpha}},\beta}=0$ unless $({{\alpha}},\beta)=({{s}},{{\ell}})$. In that case, $B_{{{s}},{{\ell}}}=1$. We have that $B_{i,{{s}}}>0$ if and only if $i={{s}}+r$, in which case 