Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 302 / 2 / i

The Pauli matrix commutator identity givesfor the anti-Hermitian basis of the SU(2) Lie algebra. In the ordered basis $(T^1,T^2,T^3)$ the Adjoint representation matrices, with columns recording the images of basis elements, areTheir squared matrix traces are $-2$, and the mixed products have zero trace. Therefore the Killing form of the SU(2) Lie algebra isIt is negative definite and nondegenerate, so this real Lie algebra is semisimple. It is also simple: the displayed bracket is the three-dimensional cross product; if an ideal of a Lie algebra contains nonzero $x$, the vectors $[y,x]$ span the plane perpendicular to $x$, and together with $x$ span the whole algebra. This confirms that the simplicity implication applies with a nonzero Killing form.