Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 102 / 1 / f

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

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