Casimir invariant C = S² = S_x² + S_y² + S_z² of su(2); eigenvalue ℏ²·s(s+1) on spin-s irrep

Casimir invariant of SU(2) spin algebra. Setup: for the su(2) Lie algebra generated by {S^x, S^y, S^z}, the quadratic Casimir operator C = S² = S_x² + S_y² + S_z² commutes with every generator ([C, S^a] = 0 for a ∈ {x, y, z}), so by…