Isomorphism between so5 and sp4 (${\mathfrak{so}}_5(\mathbb{C})\cong {\mathfrak{sp}}_4(\mathbb{C})$)

The root systems $B_2$ and $C_2$ are isomorphic after interchanging long and short simple-root labels. The classification of complex simple Lie algebras therefore gives ${\mathfrak{so}}_5(\mathbb{C})\cong {\mathfrak{sp}}_4(\mathbb{C})$.