 A2 root system

The $A_2$ root system has equal-length simple roots ${\alpha }_1,{\alpha }_2$ with angle $2\pi /3$, and roots $\pm {\alpha }_1$, $\pm {\alpha }_2$, $\pm ({\alpha }_1+{\alpha }_2)$. Its Cartan matrix is $\left(\begin{array}{cc}2& -1\\ -1& 2\end{array}\right)$. The corresponding complex simple Lie algebra is ${\mathfrak{sl}}_3(\mathbb{C})$, of dimension eight.