= Solution
Choose the maximal torus
$$
T=\{\operatorname{diag}(t_1,\ldots,t_n,t_1^{-1},\ldots,t_n^{-1})\}
$$
and write $\varepsilon_i(t)=t_i$. Then
$$
\mathfrak{sp}_{2n}=\mathfrak t\oplus\bigoplus_{\alpha\in\Phi}\mathfrak g_\alpha,
$$
where the type-$C_n$ root system is
$$
\Phi=\{\mathord\pm\varepsilon_i\mathbin\pm\varepsilon_j:i<j\}
\cup\{\mathord\pm2\varepsilon_i:1\leq i\leq n\}.
$$
Here $X^*(T)=\bigoplus_i\mathbb Z\varepsilon_i$ and $X_*(T)=\bigoplus_i\mathbb Z\varepsilon_i^\vee$. The <root datum> has
$$
(\varepsilon_i-\varepsilon_j)^\vee=\varepsilon_i^\vee-\varepsilon_j^\vee,
\quad
(\varepsilon_i+\varepsilon_j)^\vee=\varepsilon_i^\vee+\varepsilon_j^\vee,
\quad
(2\varepsilon_i)^\vee=\varepsilon_i^\vee,
$$
with the corresponding negatives.
Solved by gpt-5.6-sol high.
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