Pauli multiplication identity (${\sigma }_j{\sigma }_k={\delta }_{jk}I+i{\sum }_{\ell }{ϵ}_{jk\ell }{\sigma }_{\ell }$)

Direct multiplication of the three Pauli matrices gives $X^2=Y^2=Z^2=I$, $XY=iZ$ and their cyclic permutations, with reversed products changing sign. Consequently $(\mathbf{a}\cdot \mathbf{\sigma })(\mathbf{b}\cdot \mathbf{\sigma })=(\mathbf{a}\cdot \mathbf{b})I+i(\mathbf{a}\times \mathbf{b})\cdot \mathbf{\sigma }$. In particular $(\mathbf{s}\cdot \mathbf{\sigma }{)}^2=|\mathbf{s}{|}^{2}I$, which gives the eigenvalues $(1\pm |\mathbf{s}|)/2$ of a qubit density matrix.