= Solution
The center consists of scalar symplectic transformations:
$$
Z_G=\mu_2=\operatorname{Spec}k[z,z^{-1}]/(z^2-1).
$$
This description retains the nonreduced center in characteristic two. Since $Z_G$ is a finite central subgroup scheme, the invariant ring $k[G]^{Z_G}$ is finitely generated and
$$
\operatorname{PSp}(V)=G/Z_G=\operatorname{Spec}k[G]^{Z_G}
$$
is an <affine algebraic group>; the quotient map is finite and faithfully flat.
The quotient torus is $T/\mu_2$. Its character and cocharacter lattices are
$$
X^*(T/\mu_2)=
\left\{\sum_i a_i\varepsilon_i:\sum_i a_i\equiv0\pmod2\right\},
$$
$$
X_*(T/\mu_2)=
\mathbb Z^n+\mathbb Z\left(\frac12,\ldots,\frac12\right).
$$
The roots and coroots are the same type-$C_n$ sets written in part (ii), now regarded in these lattices. This is the adjoint root datum of type $C_n$.
Solved by gpt-5.6-sol high.
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