= Solution
Normalize the <simple roots> to $(\alpha_i,\alpha_i)=2$. The <A2 root system> has $(\alpha_1,\alpha_2)=-1$, so its <Dynkin diagram> is two nodes joined by one bond:
$$
\underset{\alpha_1}{\circ}\! -\!\underset{\alpha_2}{\circ},\qquad
\boxed{A=\begin{pmatrix}2&-1\\-1&2\end{pmatrix}.}
$$
The <fundamental weights> satisfy $(\omega_i,\alpha_j)=\delta_{ij}$. Hence the <Gram matrix> in the <fundamental weight> basis, the metric for <Dynkin label> coordinates, is
$$
\boxed{G_{ij}=(\omega_i,\omega_j)=(A^{-1})_{ij}
=\frac13\begin{pmatrix}2&1\\1&2\end{pmatrix}.}
$$
For <weights of a representation> with <Dynkin labels> $a$ and $b$, their <inner product> is $a^TGb$. In the <simple root> basis the metric is instead $A$ itself; stating the basis removes this apparent convention ambiguity.
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