Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 302 / 2 / c

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- Solution c

Solution

Since

(ad_{T_{a}})^{c}_{b} = f_{ab}^{c}

, the Killing form has components

κ_{ab} = f_{a c}^{d} f_{b d}^{c} .

This is the matrix trace of

ad_{T_{a}} ad_{T_{b}}

.

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