Past exam of the theoretical physics II course of the University of Cambridge / 2013 / Paper 1 / 5 / d

The defining matrix conditions areThe group operation is matrix multiplication. Equivalently $U=a_{0}I-i\mathbf{a}\cdot \mathbf{\sigma }$ with real $a_0,\mathbf{a}$ and $a_0^2+|\mathbf{a}{|}^2=1$, so it has three real parameters and is a compact connected simply connected Lie group. Its Hermitian spin generators obey the commutator from part (a); the corresponding Lie-algebra matrices themselves are traceless and anti-Hermitian.