corresponding columns in the matrix $B$. The mutations corresponding to the short roots again commute. We have the compound evolution $\mu^{(\Pi_<)}$ which brings us to the node $k^{(4)}$, with

1.  The cluster variables at the point $k^{(4)}$ are $\mu^{(\Pi_<)}({{\mathbf x}}[k^{(3)}])={{\mathbf x}}[k^{(4)}]=(x_1,...,x_{2r})$, with $$x_{{\alpha}}= \left\{ \begin{array}{ll} {{R}}_{{{\alpha}},2t_{{\alpha}}(k + 2)}, & {{\alpha}}\leq r \\
    {{R}}_{{{\alpha}}, 2 k + 1}, & {{\alpha}}-r\in \Pi_> \\
    {{R}}_{{{\alpha}}, 2k+3}, & {{\alpha}}-r \in \Pi_<. \end{array}\right.$$

2.  The mutation matrix at the node $k^{(4)}$ is $B^{(4)}=-B.$

Again, the proof is analogous to that of the previous lemmas.

Finally we note that at the node $k^{(4)}$, not only do the mutations $\{\mu_{{{\alpha}}+r}:{{\alpha}}\in \Pi\}$ all commute among themselves, the corresponding columns of $B^{(4)}$ describe the evolution of the $Q$-system for any ${{R}}_{{{\alpha}},i}$ with $i$ odd. Therefore the compound evolution $\mu^{(\Pi')}$ makes sense at this node. The node $k^{(5)}$ is reached from node $k^{(4)}$ by acting with $\mu^{(\Pi')}$. In fact, we find that the node $k^{(5)}$ is the node $k+1$ in ${\mathcal G}_{{\mathfrak{g}}}$:

1.  The cluster variables ${{\mathbf x}}[k^{(5)}]=\mu^{(\Pi')}({{\mathbf x}}[k^{(4)}]) =
      {{\mathbf x}}[k+1]$, where ${{\mathbf x}}[k]$ for any $k\in{{\mathbb Z}}$ was defined in equation .

2.  The mutation matrix at the node $k^{(5)}$ is $B$.

The proof is by direct calculation. 