Solution
= Solution
The second diagram is $F_4$; the arrow points from the long root $\alpha_2$ toward the short root $\alpha_3$. Thus
$$
\begin{pmatrix}
2&-1&0&0\\-1&2&-2&0\\0&-1&2&-1\\0&0&-1&2
\end{pmatrix}.
$$
= Solution
The second diagram is $F_4$; the arrow points from the long root $\alpha_2$ toward the short root $\alpha_3$. Thus
$$
\begin{pmatrix}
2&-1&0&0\\-1&2&-2&0\\0&-1&2&-1\\0&0&-1&2
\end{pmatrix}.
$$