A3 root system ($A_3$)

The $A_3$ root system is $\{e_i-e_j:1\le i\ne j\le 4\}$ in the hyperplane of ${\mathbb{R}}^4$ whose coordinates sum to zero. It has $12$ roots and simple roots $e_1-e_2,e_2-e_3,e_3-e_4$.

The Cartan matrix of type A3 is