 Matrix root basis of the even orthogonal Lie algebra

For the antidiagonal symmetric matrix $J$ in dimension $2n$, put $\bar{i}=2n+1-i$ and $H_i=E_{ii}-E_{\bar{i}\bar{i}}$. The diagonal Cartan subalgebra has coordinates ${\epsilon }_i(H)=t_i$. The root spaces of weights ${\epsilon }_i-{\epsilon }_j$ are spanned by $E_{ij}-E_{\bar{j}\bar{i}}$; those of weights ${\epsilon }_i+{\epsilon }_j$ by $E_{i\bar{j}}-E_{j\bar{i}}$, with negative-root spaces given by transposes. The condition $A_{ij}=-A_{\bar{j}\bar{i}}$ gives these generators directly and excludes roots $2{\epsilon }_i$.