Lemma: Equations and causal part

The matrix $H$ solves ${\mathsf{K}}_{\mathrm{eq}}$ in both slots and

It is even and has ghost number zero.

Proof. The auxiliary identities in (eq:H) impose $b=D\gamma$ in each slot; the other equations are the scalar wave equations. The causal identity follows from $h_{\mu }-{h_{\mu }}^{\mathsf{T}}=i{E}_{\mu }$ and slot differentiation. For the odd pair,

The reversed pair gives $-E_0$. The displayed nonzero blocks have the stated parity and ghost number.