= Solution
For $H=\operatorname{diag}(r,s,-r,-s)$, put $L_1(H)=r$, $L_2(H)=s$. Then $L_3=-L_1$, $L_4=-L_2$. The <matrix unit> identity
$$
[H,E_{ij}]=(L_i-L_j)(H)E_{ij}
$$
reads off the <root spaces> directly. The complete list is
$$
\begin{array}{c|c}
\alpha&\mathfrak g_\alpha\\ \hline
L_1-L_2&\mathbb C(E_{12}-E_{43})\\
L_2-L_1&\mathbb C(E_{21}-E_{34})\\
L_1+L_2&\mathbb C(E_{14}+E_{23})\\
-L_1-L_2&\mathbb C(E_{32}+E_{41})\\
2L_1&\mathbb CE_{13}\\
-2L_1&\mathbb CE_{31}\\
2L_2&\mathbb CE_{24}\\
-2L_2&\mathbb CE_{42}
\end{array}
$$
Thus the <C2 root system> is
$$
\boxed{\Phi=\{\pm2L_1,\ \pm2L_2,\ \pm(L_1+L_2),\ \pm(L_1-L_2)\}.}
$$
All eight <root spaces> have <dimension> one. The zero-weight space is the two-dimensional <Cartan subalgebra> $\mathbb C(E_{11}-E_{33})\oplus\mathbb C(E_{22}-E_{44})$; zero itself is not a <root>.
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