= 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 gives
$$
\boxed{\mathfrak{so}_5(\mathbb C)\cong\mathfrak{sp}_4(\mathbb C)}.
$$
This is the <Isomorphism between so5 and sp4>.
Solved by gpt-5.6-sol high.
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