Symplectic root sl2 triple ($(X_{\alpha },H_{\alpha },Y_{\alpha })$)

For the symplectic Lie algebra $\mathfrak{sp}(4)$ preserving $J=\left(\begin{array}{cc}0& I_2\\ -I_2& 0\end{array}\right)$, 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 triplesThe matrix unit commutator identity verifies $[X_{\alpha },Y_{\alpha }]=H_{\alpha }$ and the two eigenvalue relations $[H_{\alpha },X_{\alpha }]=2X_{\alpha }$, $[H_{\alpha },Y_{\alpha }]=-2Y_{\alpha }$.