= Solution
Order the <simple roots> of the <G2 root system> with $\alpha_1$ long and $\alpha_2$ short. This ordering is consistent with the supplied projection in <3/d/solution>; reversing the nodes would interchange the two fundamental <Dynkin labels>. Choose $(\alpha_1,\alpha_1)=2$, $(\alpha_2,\alpha_2)=2/3$, and $(\alpha_1,\alpha_2)=-1$. With the same row convention for the <Cartan matrix> as in <3/a/solution>,
$$
\boxed{A_{G_2}=\begin{pmatrix}2&-3\\-1&2\end{pmatrix},\qquad
\alpha_1=(2,-3),\quad\alpha_2=(-1,2).}
$$
The <positive roots>, with their <Dynkin labels>, are
$$
\begin{array}{c|c|c}
\text{root}&\text{Dynkin labels}&\text{length type}\\\hline
\alpha_2&(-1,2)&\text{short}\\
\alpha_1+\alpha_2&(1,-1)&\text{short}\\
\alpha_1+2\alpha_2&(0,1)&\text{short}\\
\alpha_1&(2,-3)&\text{long}\\
\alpha_1+3\alpha_2&(-1,3)&\text{long}\\
2\alpha_1+3\alpha_2&(1,0)&\text{long}
\end{array}
$$
For the <highest weight> $(0,1)$, a lowering chain is
$$
(0,1)\xrightarrow{-\alpha_2}(1,-1)
\xrightarrow{-\alpha_1}(-1,2)
\xrightarrow{-\alpha_2}(0,0)
\xrightarrow{-\alpha_2}(1,-2)
\xrightarrow{-\alpha_1}(-1,1)
\xrightarrow{-\alpha_2}(0,-1).
$$
The middle three terms form the length-three <weight string> for the short <simple root>. The <Weyl dimension formula>, with this long-first convention, is
$$
\dim V(a,b)=\frac{(a+1)(b+1)(a+b+2)(2a+b+3)(3a+b+4)(3a+2b+5)}{120}.
$$
It gives seven for $(0,1)$, so the chain exhausts the <weight system>, with every <weight multiplicity> equal to one:
$$
\boxed{\mathbf7=(0,1):\quad(0,0),\quad
\pm(0,1),\quad\pm(1,-1),\quad\pm(-1,2).}
$$
For $(1,0)$, observe that $2\alpha_1+3\alpha_2=(1,0)$ is the <highest root>. Thus this is the <Adjoint representation>; the twelve <root spaces> contribute the twelve nonzero <weights> once, and the rank-two <Cartan subalgebra> contributes two zero <weights>. Therefore
$$
\boxed{\mathbf{14}=(1,0):\quad
\pm(2,-3),\ \pm(-1,3),\ \pm(1,0),\
\pm(-1,2),\ \pm(1,-1),\ \pm(0,1),\ (0,0)^{[2]}.}
$$
The same <Weyl dimension formula> gives fourteen, confirming the count.
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