usually subgraphs of a larger component of $\mathbb T_n$ with the same property, and therefore our definition is not unique. There may be other nodes of $\mathbb T_n$ with cluster variables in ${\mathcal A}$ which are not in ${\mathcal G}_{{\mathfrak{g}}}$. This happens if the rank of ${{\mathfrak{g}}}$ is greater than two. We will give the example of $A_3$ to make this point clearer in the discussion.

The $Q$-systems as cluster algebras for simple ${{\mathfrak{g}}}$ {#qascluster}
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Let $C$ be a Cartan matrix of a simple Lie algebra ${{\mathfrak{g}}}$ of rank $r$, and let $B$ be the $2r \times 2r$ matrix defined in the block form: $$\label{Bmatrix}
B = \left(\begin{array}{cc} A & -C^t\\   C & 0 \end{array}\right),$$ where $A=C^t-C$. This is a skew-symmetric matrix of integers.

For each $k\in {{\mathbb Z}}$, define ${{\mathbf x}}[k] = (x_1,...,x_{2r})$, where $$\label{clustervariable}
x_{{\alpha}}= \left\{\begin{array}{ll} {{R}}_{{{\alpha}}, 2 t_{{\alpha}}k}, & {{\alpha}}\leq r \\
{{R}}_{{{\alpha}}-r,2 t_{{\alpha}}k + 1}, & r<{{\alpha}}\leq 2r \end{array}\right.$$ Here, $t_{{\alpha}}=2$ for the short roots of $B_r, ~C_r$ and $F_4$, $t_{{\alpha}}=3$ for the short root of $G_2$, and and $t_{{\alpha}}=1$ otherwise.

In our cluster algebra graph, nodes labeled by $k\in {{\mathbb Z}}$ have associated with them cluster $({{\mathbf x}}[k], B)$. We will prove the following 