it is equal to $2$. Thus, we have $$B_{{{s}}+r,{{\ell}}}^{(1)} = 0,$$ and all other elements in the long-root columns of the lower left hand corner of $B^{(1)}$ are unchanged. This gives the matrix $D$ in this quadrant, and $-D^t$ in the upper right block of $B^{(1)}$ since mutations preserve skew-symmetry.

-   Finally, elements in the lower right quadrant mutate as follows: $$\begin{aligned}
    B_{i+r,j+r}^{(1)} &=& \sum_{{{\alpha}}\in \Pi_<}
    {{\rm sgn}}(B_{i+r,{{\alpha}}})[B_{i+r,{{\alpha}}}B_{{{\alpha}},j+r}]_+ = 2A.\end{aligned}$$

By examination of the matrix $B^{(1)}$ we conclude that at the node $k^{(1)}$, the mutations $\{\mu_{{\alpha}}:~{{\alpha}}\in \Pi_>\}$ all commute among themselves.

1.  The cluster variables ${{\mathbf x}}[k^{(2)}]=(x_1,...,x_{2r})=\mu^{(\Pi_>)}({{\mathbf x}}[k^{(1)}])$ are $$x_{{{\alpha}}} = \left\{ \begin{array}{ll} {{R}}_{2 t_{{\alpha}}k + 2} & {{\alpha}}\leq r; \\
    {{R}}_{2 t_{{\alpha}}k + 1} & {{\alpha}}>r.\end{array}\right.$$

2.  The mutation matrix $B^{(2)}$ has the form $$\left(\begin{array}{cc} A & C-A \\ -C^t-A & 2A \end{array}\right).$$

Again, by inspection, the columns corresponding to long roots in the matrix $B^{(1)}$ correspond to the evolution of the $Q$-system according to Equations and . This proves the first statement in the Lemma.

We look at the evolution of the matrix $B^{(1)}$ under $\mu^{(\Pi_>)}$.

-   If ${{\alpha}}\in \Pi_>$ then $B_{i,{{\alpha}}}^{(2)} =
      -B_{i,{{\alpha}}}^{(1)}$. This accounts for the appearance of $A$ in the upper left-hand block of $B^{(2)}$ 