D5 root system ($D_5$)

The root system $D_5$ consists of the forty roots $\pm e_i\pm e_j$, $1\le i<j\le 5$, in an orthonormal basis. One choice of simple roots is $e_1-e_2,e_2-e_3,e_3-e_4,e_4-e_5,e_4+e_5$. The Dynkin diagram has a fork at node $3$ with ends $4$ and $5$. Its Weyl group permutes coordinates and changes an even number of signs. The associated complex simple Lie algebra is $\mathfrak{so}(10,\mathbb{C})$. Its two half-spin representations have dimension $16$, with highest weights $\frac{1}{2}(e_1+e_2+e_3+e_4\mp e_5)$.