 Pauli error criterion for a stabilizer code

Let $S$ exclude $-I$, let $C(S)$ be its Pauli centralizer, and let $\mathcal{Z}=\{\pm I,\pm iI\}$. A set of Pauli errors is jointly correctable exactly when every product $E_a^{†}{E}_b$ belongs to $\mathcal{Z}S$ or lies outside $C(S)$. Outside the centralizer its code compression is zero by anticommutation; in $\mathcal{Z}S$ it acts as a scalar. A centralizing operator outside $\mathcal{Z}S$ has zero trace on the code but is unitary there, so cannot be scalar. This gives precisely the Knill--Laflamme condition. Scalar phases must be retained when formulating the group criterion.