= 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 $\varepsilon_i(H)=t_i$. The <root spaces> of weights $\varepsilon_i-\varepsilon_j$ are spanned by $E_{ij}-E_{\bar j\bar i}$; those of weights $\varepsilon_i+\varepsilon_j$ 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 $2\varepsilon_i$.
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