and for the columns corresponding to the long roots in the lower left-hand quadrant of $B^{(2)}$.

-   If ${{\alpha}}\in \Pi_<$ and $i>r$ then $$B_{i,{{\alpha}}}^{(2)} = B_{i,{{\alpha}}}^{(1)} + \sum_{\beta\in \Pi_>}
    {{\rm sgn}}(B_{i,\beta}^{(1)}) [B_{i,\beta}^{(1)}B_{\beta,{{\alpha}}}^{(1)}]_+$$ The summation has a contribution only when $(\beta,{{\alpha}})=({{\ell}},{{s}})$ in which case $B_{{{\ell}},{{s}}}^{(1)}=1$. Then the only non-vanishing contribution occurs if $i=\beta+r$, in which case $B_{\beta+r,\beta}^{(1)}=2$ and $B_{{{\ell}}+r,{{s}}} = 3$. This gives the matrix in the lower left quadrant of $B^{(2)}$.

-   Finally we have $$B^{(2)}_{i+r,j+r} = 2A_{i,j} + \sum_{{{\alpha}}\in
      \Pi_>}{{\rm sgn}}(B_{i+r,{{\alpha}}}^{(1)})[B_{i+r,{{\alpha}}}^{(1)} B_{{{\alpha}},j+r}^{(1)}]_+
    = 2A_{ij}.$$

At this point, it is evident that the mutations $\{\mu_{{{\alpha}}+r}:~{{\alpha}}\in
\Pi_<\}$ commute among themselves, due to the form of the matrix $2A$ appearing in the lower right quadrant of $B^{(2)}$. Moreover, the corresponding columns of $B^{(2)}$ give the $Q$-system evolution for ${{R}}_{{{\alpha}},2k+1}\mapsto {{R}}_{{{\alpha}},2k+3}$ if ${{\alpha}}$ is short. As a result we have

1.  The cluster variables at $k^{(3)}$ are $\mu^{(\Pi_<')}({{\mathbf x}}[k^{(2)}])={{\mathbf x}}[k^{(3)}]=(x_1,...,x_i)$ where $$x_{{{\alpha}}+r} = {{R}}_{{{\alpha}},4k+3},\qquad {{\alpha}}\in \Pi_<,$$ and the other cluster variables are unchanged from ${{\mathbf x}}[k^{(2)}]$.

2.  The mutation matrix $B^{(3)}=-B^{(1)}$

The proof is by a calculation analogous to the previous two lemmas.

The columns $\{{{\alpha}}:~{{\alpha}}\in \Pi_<\}$ of $-B^{(1)}$ describe the mutations corresponding to the evolutions ${{R}}_{{{\alpha}},4k+2}\to{{R}}_{{{\alpha}},4k+4}$ as they are identical to the 