= Symplectic root sl2 triple
{title2=$(X_\alpha,H_\alpha,Y_\alpha)$}
For the <symplectic Lie algebra> $\mathfrak{sp}(4)$ preserving $J=\begin{pmatrix}0&I_2\\-I_2&0\end{pmatrix}$, write $h_1=E_{11}-E_{33}$ and $h_2=E_{22}-E_{44}$. Its positive <roots of a root system> have <sl2 subalgebra associated with a root> triples
$$
\begin{array}{c|c|c|c}
\alpha&X_\alpha&H_\alpha&Y_\alpha\\\hline
\varepsilon_1-\varepsilon_2&E_{12}-E_{43}&h_1-h_2&E_{21}-E_{34}\\
\varepsilon_1+\varepsilon_2&E_{14}+E_{23}&h_1+h_2&E_{41}+E_{32}\\
2\varepsilon_1&E_{13}&h_1&E_{31}\\
2\varepsilon_2&E_{24}&h_2&E_{42}
\end{array}
$$
The <matrix unit> commutator identity verifies $[X_\alpha,Y_\alpha]=H_\alpha$ and the two eigenvalue relations $[H_\alpha,X_\alpha]=2X_\alpha$, $[H_\alpha,Y_\alpha]=-2Y_\alpha$.
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