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>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 102 / 3 / e / Solution

{c}
{title2=$\mathfrak{so}_5(\mathbb C)\cong\mathfrak{sp}_4(\mathbb C)$}

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 $\mathfrak{so}_5(\mathbb C)\cong\mathfrak{sp}_4(\mathbb C)$.


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