The <Killing form> is $\kappa(X,Y)=\operatorname{Tr}(\operatorname{ad}_X\operatorname{ad}_Y)$. From part (a),
$$
\begin{aligned}
\kappa([X,Y],Z)
&=\operatorname{Tr}([\operatorname{ad}_X,\operatorname{ad}_Y]\operatorname{ad}_Z)\\
&=\operatorname{Tr}(\operatorname{ad}_X[\operatorname{ad}_Y,\operatorname{ad}_Z])\\
&=\kappa(X,[Y,Z]),
\end{aligned}
$$
where the middle equality uses cyclicity of the <trace>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 302 / 3 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-302/3/b/solution