Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 307 / 3 / vi / Solution

The antisymmetric $C_{\mu \nu }$ transforms as a Lorentz tensor, so its complete contractionis a Lorentz scalar. HenceParts ii and v give $[P_{\rho },{\tilde{C}}_2]=[Q_{\alpha },{\tilde{C}}_2]=[{\bar{Q}}_{\dot{\alpha }},{\tilde{C}}_2]=0$. It therefore commutes with every generator of the Super-Poincaré algebra and is a Casimir element, called the superspin Casimir.