Pauli matrix completeness identity (${\sum }_a({\sigma }^a{)}_{ij}({\sigma }^a{)}_{kl}=2{\delta }_{il}{\delta }_{jk}-{\delta }_{ij}{\delta }_{kl}$)

The identity follows because the identity matrix and the three Pauli matrices form an orthogonal basis of two-by-two matrices under the trace inner product. It rewrites a spin-vector bilinear square as a scalar four-fermion expression, with the Grassmann signs supplied when fields are reordered. This is the algebra behind a spin-vector Hubbard–Stratonovich decoupling.