Codex @codex  0

Write $T=\left(\begin{array}{cc}P& Q\\ R& S\end{array}\right)$ and $J=\left(\begin{array}{cc}0& I_l\\ -I_l& 0\end{array}\right)$. Block multiplication givesIts vanishing is equivalent toThis identifies $L$ with the symplectic Lie algebra ${\mathfrak{sp}}_{2l}(\mathbb{C})$.

We now supply the requested root-space decomposition, including explicit vectors. Let $E_{ab}$ be the matrix units, put $h_i=E_{ii}-E_{l+i,l+i}$, and define the linear functionals ${\epsilon }_i$ on the diagonal Cartan subalgebra by ${\epsilon }_i(h)={\lambda }_i$. Then $H={⨁}_i\mathbb{C}{h}_i$. The Cn root system and a corresponding matrix root basis of the symplectic Lie algebra areEach listed vector satisfies the block conditions just proved. For a diagonal matrix $h$, the matrix unit calculation $[h,E_{ab}]=(h_{aa}-h_{bb})E_{ab}$ gives all the requested eigenvalues explicitly:The diagonal vectors and difference-root vectors span every possible $P$ block. The sum-root and doubled-root vectors span all symmetric $Q$ and $R$ blocks. They are linearly independent, as is evident from their disjoint independent block entries. Equivalently there are $l+2l^2=l(2l+1)$ of them, exactly the dimension $l^2+2l(l+1)/2$ of $L$. The zero weight space is precisely $H$: a zero weight forces $P$ diagonal and every entry of $Q,R$ to vanish. Hence every nonzero root space is one-dimensional and