= Solution
Choose the <root basis>
$$
\boxed{\alpha_1=\varepsilon_1-\varepsilon_2,\qquad
\alpha_2=\varepsilon_2-\varepsilon_3,\qquad
\alpha_3=\varepsilon_2+\varepsilon_3.}
$$
In the usual invariant <inner product>, each <simple root> has squared length $2$, while $(\alpha_1,\alpha_2)=(\alpha_1,\alpha_3)=-1$ and $(\alpha_2,\alpha_3)=0$. The labeled <Dynkin diagram> is therefore the following chain, with a single bond between adjacent nodes:
$$
\underset{\alpha_2}{\circ}\;\text{---}\;\underset{\alpha_1}{\circ}\;\text{---}\;\underset{\alpha_3}{\circ}.
$$
It is type $D_3$, equivalently the <A3 root system>. To check the <root basis> explicitly, the corresponding <positive roots> are $\alpha_1,\alpha_2,\alpha_3,\alpha_1+\alpha_2,\alpha_1+\alpha_3,\alpha_1+\alpha_2+\alpha_3$; these and their negatives are all twelve roots.
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