= Solution
Write an element of the diagonal <Cartan subalgebra> as
$$
H(t)=\operatorname{diag}(t_1,t_2,t_3,-t_1,-t_2,-t_3),\qquad
\varepsilon_i(H(t))=t_i.
$$
The <root system> of this <Special orthogonal Lie algebra> is
$$
\boxed{\Phi=\{\pm\varepsilon_i\pm\varepsilon_j:1\leq i<j\leq3\}.}
$$
Thus the maps requested are $H(t)\mapsto\pm t_i\pm t_j$, with the two signs independent. This is the <D3 root system>, with twelve roots.
For a direct check, the defining matrix identity gives the block form $X=\begin{pmatrix}A&B\\C&-A^T\end{pmatrix}$ with $B^T=-B$ and $C^T=-C$. If $E_{ij}$ denotes a <matrix unit>, the <root space> for $\varepsilon_i-\varepsilon_j$ is spanned by $E_{ij}-E_{j+3,i+3}$; those for $\varepsilon_i+\varepsilon_j$ and $-\varepsilon_i-\varepsilon_j$ are spanned respectively by $E_{i,j+3}-E_{j,i+3}$ and $E_{i+3,j}-E_{j+3,i}$. Commuting these with $H(t)$ gives exactly the displayed maps.
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