= Cartan matrix convention for C3
{c}
{title2=$A_{ij}=\langle\alpha_i,\alpha_j^\vee\rangle$}
For the <C3 root system>, take $\alpha_1=e_1-e_2$, $\alpha_2=e_2-e_3$, $\alpha_3=2e_3$. With columns indexing <coroots>, the <Cartan matrix> is
$$
A_{ij}=\frac{2(\alpha_i,\alpha_j)}{(\alpha_j,\alpha_j)},\qquad A=\begin{pmatrix}2&-1&0\\-1&2&-1\\0&-2&2\end{pmatrix}.
$$
The convention with rows indexing coroots gives $A^T$. Both describe a double bond between the short root $\alpha_2$ and long root $\alpha_3$, with arrow toward $\alpha_2$.
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