D3 root system ($D_3$)

The $D_3$ roots are $\pm {\epsilon }_i\pm {\epsilon }_j$ for $1\le i<j\le 3$. The simple rootsform the three-node chain ${\alpha }_2-{\alpha }_1-{\alpha }_3$, so $D_3\cong A_3$.

The root-system isomorphism $D_3\cong A_3$ and the classification of complex simple Lie algebras give ${\mathfrak{so}}_6(\mathbb{C})\cong {\mathfrak{sl}}_4(\mathbb{C})$.