= Trace form on scalar-plus-skew-Hermitian two-by-two matrices
{title2=$C=aI+i\mathbf v\cdot\boldsymbol\sigma,\quad-\tfrac12\operatorname{tr}(CD)=-ab+\mathbf v\cdot\mathbf w$}
The real matrix space defined by $C+C^*=\operatorname{tr}(C)I$ consists exactly of $C=aI+i\mathbf v\cdot\boldsymbol\sigma$, with $a$ and the three components of $\mathbf v$ real. The <Pauli matrices> have $\operatorname{tr}(\sigma_i\sigma_j)=2\delta_{ij}$, giving the displayed real symmetric <bilinear form> of signature $(3,1)$. The real isomorphism $C\mapsto aI-\mathbf v\cdot\boldsymbol\sigma$ to <Hermitian matrices> preserves this form when the latter have the polarized negative-determinant form $[\operatorname{tr}(AB)-\operatorname{tr}A\operatorname{tr}B]/2$.
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