-   $\Pi_>$: The subset in $I_r$ corresponding to the long roots of ${{\mathfrak{g}}}$;

-   $\Pi_<'=\{{{\alpha}}+r ~|~ {{\alpha}}\in \Pi_<\}$;

-   $\Pi_>'=\{{{\alpha}}+r ~|~ {{\alpha}}\in \Pi_>\}$;

The graph ${\mathcal G}_{{\mathfrak{g}}}$ is obtained from the sequence

(0,0)(60,10) (5,5)(55,5) (5,5)(10,0)[6]{}[(0,0)[.7]{}]{} (5,3)[$k$]{} (15,3)[$k^{(1)}$]{} (25,3)[$k^{(2)}$]{} (35,3)[$k^{(3)}$]{} (45,3)[$k^{(4)}$]{} (55,3)[$k+1$]{} (3,5)[$\cdots$]{} (58,5)[$\cdots$]{} (10,7)[${\Pi_<}$]{} (20,7)[${\Pi_>}$]{} (30,7)[${\Pi_<'}$]{} (40,7)[${\Pi_<}$]{} (50,7)[${\Pi'}$]{}

Here, the cluster at the node labeled $k$ is ${{\mathbf x}}[k]$ as in with mutation matrix $B$ as in . All the compound mutations are products of mutually commuting mutations, and the cluster variables at each node are in ${\mathcal A}$.

To prove this Theorem, we need only list the cluster variables $({{\mathbf x}}[k^{(i)}],B^{(i)})$ at $k^{(i)}$ – at intermediate nodes, the cluster variables are a mixture of cluster variables from $k^{(i-1)}$ and $k^{(i)}$. Note that in this subsection, we read the superscript of $k$ modulo 5, so $k^{(0)}:=k$ etc.

We then observe that the relevant columns of $B^{(i)}$ describe $Q$-system evolutions when applied to the relevant entries of ${{\mathbf x}}[k^{(i)}]$. We use the fact that when $\mu_i$ and $\mu_j$ commute, the $j$th column of $\mu_i(B)$ is equal to the $j$th column of $B$.

We recall that at the node $k$, we have ${{\mathbf x}}[k] = (x_1,...,x_{2r})$, where $$x_{{\alpha}}= \left\{\begin{array}{ll} {{R}}_{{{\alpha}},2t_{{\alpha}}k},& {{\alpha}}\in I_r;\\
