{{R}}_{{{\alpha}},2t_{{\alpha}}k + 1}, & \hbox{otherwise}.\end{array}\right.$$ Moreover, according to , at the node $k$ we have $$B = \left(\begin{array}{cc} A & -C^t\\   C & 0 \end{array}\right).$$ Here, the submatrix $A$ is no longer the zero matrix. Instead, it has zeros everywhere except for two entries, $A_{{{\ell}},{{s}}}=-1$ and $A_{{{s}},{{\ell}}}=1$. Thus, the mutations $\{\mu_{{\alpha}}|\ {{\alpha}}\in \Pi_<\} $ commute among themselves, as do the all mutations $\{\mu_{i} |\ i\in \Pi'\}$.

It is clear from the form of the $Q$-system (see Equations and ) that the mutations $\{ \mu_{{\alpha}}|~{{\alpha}}\in \Pi_<\}$ describe the evolutions $\mu_{{\alpha}}: {{R}}_{{{\alpha}},2k}\mapsto {{R}}_{{{\alpha}},2k+2}$ (${{\alpha}}\in
\Pi_<$). This is the same evolution as in the simply-laced case if ${{\alpha}}\neq {{s}}$, and for ${{\alpha}}={{s}}$, it follows from .

Thus, edges labeled by ${{\alpha}}\in \Pi_<$ connected to node $k$ lead to nodes in which all the cluster variables are in ${\mathcal A}$. The compound mutation $\mu^{(\Pi_<)}$ leads to the node $k^{(1)}$.

(We also note that the mutations columns $r+1,...,2r$ describe the $Q$-system evolutions ${{R}}_{{{\alpha}},2k+1}\mapsto {{R}}_{{{\alpha}},2k-1}$, from Equations , and ; However due to the periodic structure of the graph, we do not need to consider this evolution separately).

1.  The cluster variables at the node $k^{(1)}$ are $
    {{\mathbf x}}[k^{(1)}] = (x_1,...,x_{2r})=\mu^{(\Pi_<)}({{\mathbf x}}[k])$ with $$x_{{\alpha}}=\left\{ \begin{array}{ll}
     