Pauli matrix multiplication law

The Pauli matrices satisfyConsequently each squares to the identity, distinct Pauli matrices anticommute, and their products span $M_2(\mathbb{C})$.

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.

For vectors $\mathbf{a},\mathbf{b}\in {\mathbb{R}}^3$, the Pauli matrix multiplication law gives