$$\mu^{(I_r)}=\mu_1\circ \cdots \circ \mu_r$$ and denote the node reached by this compound mutation acting on the cluster variables at $k$ by $k'$. Then the mutation matrix $B'=\mu^{(I_r)}(B)$ is $$B'=-B = \left(\begin{array}{rr} 0 & C \\ -C & 0\end{array}\right).$$ and $$\mu^{(I_r)}({{\mathbf x}}[k]) =
    ({{R}}_{1,2k+2},...,{{R}}_{r,2k+2},{{R}}_{1,2k+1},...,{{R}}_{r,2k+1}) = {{\mathbf x}}[k'].$$

4.  At the point $k'$, the mutations $\mu_{r+1 },...,\mu_{2r}$ commute among themselves, the evolution according to these mutations corresponds to the $Q$-system evolution ${{R}}_{{{\alpha}},2k+1}\mapsto
      {{R}}_{{{\alpha}},2k+3}$.

5.  The compound mutation $$\mu^{(I_r')} = \mu_{r+1}\circ\cdots\circ \mu_{2r}$$ maps the seed at $k'$ to the seed at $k+1$ with cluster variables ${{\mathbf x}}[k+1]$ and mutation matrix $B$.

This theorem is proved by direct calculation.

Let us illustrate some examples of graphs which correspond to ${\mathcal G}_{{\mathfrak{g}}}$.

The graph ${\mathcal G}_{{{\mathfrak{sl}}}_2}$ is linear:

(0,0)(50,10) (5,5)(45,5) (5,5)(10,0)[5]{}[(0,0)[.7]{}]{} (5,3)[$(k-1)'$]{} (15,3)[$k$]{} (25,3)[$k'$]{} (35,3)[$k+1$]{} (45,3)[$(k+1)'$]{} (3,5)[$\cdots$]{} (48,5)[$\cdots$]{} (10,6)[2]{} (20,6)[1]{} (30,6)[2]{} (40,6)[1]{}

The cluster variables at the primed and unprimed nodes are described by ${{\mathbf x}}[k]=({{R}}_{1,2k},{{R}}_{1,2k+1})$ and ${{\mathbf x}}[k']=({{R}}_{1,2k+2},{{R}}_{1,2k+1})$ of Theorem \[simple\]. The numbers above each edge correspond to the mutations $\mu_1$ and $\mu_2$. This is the full tree $\mathbb T_2$.

The graph ${\mathcal G}_{{{\mathfrak{sl}}}_3}$ is slightly more interesting:

(0,0)(80,30) (0,0)(20,0)[4]{}[(0,10)(10,20)(20,10)(10,0)(0,10)]{} (0,10)(20,0)[5]{}[(0,0)[.7]{}]{} (10,0)(20,0)[4]{}[(0,0)[.7]{}]{} (10,20)(20,0)[4]{}[(0,0)[.7]{}]{} (-2,7)[$(k-1)'$]{} (20,7)[$k$]{} (40,7)[$k'$]{} (60,7)[$k+1$]{} (82,7)[$(k+1)'$]{} (-3,10)[$\cdots$]{} (83,10)[$\cdots$]{} (6,6)[4]{} (26,6)[2]{} (46,6)[4]{} (66,6)[2]{} 