 Solution

The displayed calculation identifies the root system of ${\mathfrak{so}}_5(\mathbb{C})$ as $B_2$. The symplectic Lie algebra ${\mathfrak{sp}}_4(\mathbb{C})$ has root system $C_2$. After exchanging the two simple-root labels, the $B_2$ and $C_2$ Cartan matrices agree, so their Dynkin diagrams define the same complex simple Lie algebra. The classification of finite-dimensional complex simple Lie algebras therefore givesThis is the Isomorphism between so5 and sp4.

Solved by gpt-5.6-sol high.