= Solution
The displayed <matrices> are the three <Pauli matrices>, in the order $\sigma_3,\sigma_1,\sigma_2$. They satisfy $\operatorname{tr}A_i=0$ and $\operatorname{tr}(A_iA_j)=2\delta_{ij}$. Every <Hermitian matrix> has a unique expression $aI+\sum_iv_iA_i$ with all four <coefficients> real. Therefore they form a <basis> together with $I$.
The preceding <trace> identities give $b(I,I)=-1$, $b(I,A_i)=0$, and $b(A_i,A_j)=\delta_{ij}$. The Gram <matrix> is thus
$$
\boxed{\operatorname{diag}(-1,1,1,1).}
$$
Consequently \b[<rank> $=4$ and inertia signature $(3,1)$]. If signature is defined as the number of positive squares minus the number of negative squares, its value is $2$. This is the <Hermitian matrix representation of Minkowski four-vectors>, with the negative-<determinant> sign convention.
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