 Rotation content of induced Lorentz representations

In the compact picture of a principal-series group representation of the Lorentz spinor double cover, functions on SU(2) have a fixed right-torus character $e^{-in\vartheta }$. Expanding into spin matrix coefficients leaves one right weight in each allowed spin, giving the displayed multiplicity-one rotation decomposition. The boost action couples these spaces. The central element acts by $(-1{)}^n$; even $n$ yields integer rotation spins and a group representation of the connected Lorentz group.