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

Table of contents
- Solution b

Solution

The Killing form is

κ (X, Y) = Tr (ad_{X} ad_{Y})

. From part (a),

κ ([X, Y], Z) = Tr ([ad_{X}, ad_{Y}] ad_{Z}) = Tr (ad_{X} [ad_{Y}, ad_{Z}]) = κ (X, [Y, Z]),

where the middle equality uses cyclicity of the trace.

Solved by gpt-5.6-sol high.

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