= Solution
A simply-laced positive-definite <Coxeter graph> has no multiple edges and has positive-definite symmetric Cartan matrix, with diagonal entries $2$ and entry $-1$ precisely across an edge. Its connected components are exactly the <ADE classification>
$$
\boxed{A_n\ (n\geq1),\quad D_n\ (n\geq4),\quad E_6,E_7,E_8.}
$$
The graph underlying $1\to2$ is $A_2$. For simple roots $\alpha_1,\alpha_2$ with $(\alpha_i,\alpha_i)=2$ and $(\alpha_1,\alpha_2)=-1$, its roots are
$$
\boxed{\pm\alpha_1,\quad\pm\alpha_2,\quad\pm(\alpha_1+\alpha_2).}
$$
There is one base for each Weyl chamber, hence six bases. The <Weyl group> is generated by the two root reflections with
$$
s_1^2=s_2^2=1,\qquad(s_1s_2)^3=1,
$$
so $W\simeq S_3$. Relative to a chosen base, the two <Coxeter elements> are $s_1s_2$ and $s_2s_1$; both have order three.
A representation of the quiver is one linear map $f:V_1\to V_2$. Choosing bases that put $f$ in rank normal form decomposes it into copies of
$$
(k\to0),\qquad(0\to k),\qquad(k\xrightarrow{1}k).
$$
These are indecomposable and have dimension vectors $(1,0)$, $(0,1)$, and $(1,1)$, the three positive roots of $A_2$. More generally, <Gabriel theorem> says that for any orientation of a simply-laced positive-definite Dynkin graph, indecomposable quiver representations are in bijection with its positive roots. The finite ADE root system therefore gives finitely many indecomposables.
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