The matrix is the <Cartan matrix of type A3>. The positive roots are
$$
\alpha_1,
\quad\alpha_2,
\quad\alpha_3,
\quad\alpha_1+\alpha_2,
\quad\alpha_2+\alpha_3,
\quad\alpha_1+\alpha_2+\alpha_3.
$$
Together with their negatives they form
$$
\Phi=\{\pm\alpha_1,\pm\alpha_2,\pm\alpha_3,
\pm(\alpha_1+\alpha_2),\pm(\alpha_2+\alpha_3),
\pm(\alpha_1+\alpha_2+\alpha_3)\},
$$
so $|\Phi|=12$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 302 / 3 / c / Solution

{c}
{title2=$A_3$ Cartan matrix}

The Cartan matrix of type A3 is
$$
\begin{pmatrix}2&-1&0\\-1&2&-1\\0&-1&2\end{pmatrix}.
$$


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