Past exam of the groups and representations course of the University of Oxford / 2018 / 3 / c / Solution

Choose the following Pauli matrix realization, with its spinor basis fixed for the later Dirac adjoint:The Pauli matrix multiplication rule verifiesExtending linearly therefore gives a two-dimensional Lie algebra representation. For an irreducible representation, it suffices to rule out an invariant line. Any such line would have to be an eigenline of $r(K_1)$, whose two eigenvalues are distinct. Those two coordinate lines are interchanged by $r(K_2)$, so neither is invariant. There is no proper nonzero invariant subspace.

This is a spinorial representation rather than a single-valued two-dimensional representation of the identity component of the specified orthogonal group:The mismatch explicitly exhibits the need for the spin double cover. It causes no obstruction to the requested infinitesimal construction.