A4 root system ($A_4$)

The root system $\{e_i-e_j:1\le i\ne j\le 5\}$ belongs to the complex special linear Lie algebra ${\mathfrak{sl}}_5(\mathbb{C})$. Its four consecutive-difference simple roots give a four-node An Dynkin diagram. The compact real form is the Lie algebra of $SU(5)$.