= Solution
The diagonal <Cartan subalgebra> is
$$
\mathfrak h=\{H(a,d)=\operatorname{diag}(a,d,-a,-d,0):a,d\in\mathbb C\}.
$$
Put $\varepsilon_1=L_1$ and $\varepsilon_2=L_2$; then $L_3=-\varepsilon_1$, $L_4=-\varepsilon_2$ and $L_5=0$ on $\mathfrak h$. For the <matrix unit> $E_{ij}$, $[H,E_{ij}]=(L_i-L_j)(H)E_{ij}$. Applying this to the general matrix in part (i) gives the following one-dimensional <root spaces> and chosen root vectors:
$$
\begin{array}{c|l}
\alpha&X_\alpha\\\hline
\varepsilon_1-\varepsilon_2&E_{12}-E_{43}\\
-\varepsilon_1+\varepsilon_2&E_{21}-E_{34}\\
\varepsilon_1+\varepsilon_2&E_{14}-E_{23}\\
-\varepsilon_1-\varepsilon_2&E_{32}-E_{41}\\
\varepsilon_1&E_{15}-E_{53}\\
-\varepsilon_1&E_{35}-E_{51}\\
\varepsilon_2&E_{25}-E_{54}\\
-\varepsilon_2&E_{45}-E_{52}
\end{array}
$$
Each row satisfies $[H,X_\alpha]=\alpha(H)X_\alpha$ and belongs to the <Special orthogonal Lie algebra>. The two diagonal parameters and eight root vectors span the ten-parameter matrix from part (i), giving the explicit <root-space decomposition>
$$
\boxed{\mathfrak{so}_5=\mathfrak h\oplus\bigoplus_{\alpha\in R}\mathbb C X_\alpha,\qquad R=\{\pm\varepsilon_1,\pm\varepsilon_2,\pm\varepsilon_1\pm\varepsilon_2\}.}
$$
This is the <B2 root system>, with short roots on the coordinate axes and long roots on the diagonals.
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