Proposition: Auxiliary Green matrix

For either retarded or advanced support,

\mathcal G_{rb}^{\mathrm R/\mathrm A}
  &=
  \begin{pmatrix}
   \Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}
       &\tfrac12\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}\mathsf C^*\\
   \tfrac12\mathsf C\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}
       &\tfrac14\mathsf C\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}
                                      \mathsf C^*-\tfrac12I
  \end{pmatrix}
 \nonumber\\
  &=
  \begin{pmatrix}
     -4\mathsf P_{\mathrm{tr}}\mathsf E_0^{\mathrm R/\mathrm A}
          &K\mathsf E_0^{\mathrm R/\mathrm A}\\
     -2\mathsf C\mathsf P_{\mathrm{tr}}\mathsf E_0^{\mathrm R/\mathrm A}&0
  \end{pmatrix}.
 \label{eq:Ghb}

Both inverse identities hold on compact tests. The operators have the indicated causal support.

Proof. The second equation gives $b=(\mathsf{C}r-j)/2$. Substitution into the first gives

Apply the appropriate metric Green inverse and substitute back for $b$. This yields the first matrix.

The identities

follow from (eq:Cstar) and $\square {{\mathsf{E}}_0}^{\mathrm{R}/\mathrm{A}}=1$. They give the second matrix. Substitution verifies the right inverse; applying the same elimination to a compact field pair verifies the left inverse. Uniqueness reduces to that of the free metric wave equation. Derivatives and the cancelled algebraic block preserve causal support.