 Dn root system

For the complexified Special orthogonal Lie algebra ${\mathfrak{so}}_{2n}$, $n\ge 2$, choose positive roots of a root system $e_i-e_j$ and $e_i+e_j$ for $i<j$. Its Weyl vector is $\rho =(n-1,n-2,\dots ,0)$. Dominant coordinates satisfy ${\lambda }_1\ge \cdots \ge {\lambda }_{n-1}\ge |{\lambda }_n|$. The Weyl dimension formula becomes ${\prod }_{i<j}(({\lambda }_i+{\rho }_i{)}^2-({\lambda }_j+{\rho }_j{)}^2)/({\rho }_i^2-{\rho }_j^2)$. The case $D_2$ is two independent A1 root systems, while $D_3$ is isomorphic to the A3 root system.