 Solution

For ${\mathfrak{sl}}_3(\mathbb{C})$, the Killing form of the special linear Lie algebra isFor $H_1=E_{11}-E_{22}$ and $H_2=E_{22}-E_{33}$, its matrix on the Cartan part iswhose determinant is $108$. Each $E_{ij}$ pairs only with $E_{ji}$, with value $6$. In the stated ordering, the remaining block iswhose determinant is $-6^6$. Therefore

Solved by gpt-5.6-sol high.