Proposition: Full auxiliary inverse

The unique retarded and advanced inverses are

G^{\mathrm{aux}}^\pm=
 \begin{pmatrix}
 G^{\mathrm{gr}}^\pm&G^{\mathrm{gr}}^\pm D^*\\
 DG^{\mathrm{gr}}^\pm&DG^{\mathrm{gr}}^\pm D^*-\frac12\operatorname{id}
 \end{pmatrix}
 =
 \begin{pmatrix}
 -\mathcal P\mathsf E_0^\pm&\mathcal R\mathsf E_0^\pm\\
 D(-\mathcal P\mathsf E_0^\pm)&0
 \end{pmatrix}.
 \label{eq:Gaux}

The identities in the second matrix hold on compact interior sources.

Proof. The auxiliary equation gives $b=D\gamma -j/2$. Substitution reduces the metric equation to

The handed-off identity is $A+2D^{\ast }D=-\square \mathcal{P}$. Equation (eq:Ggr) therefore gives the first matrix and uniqueness.

For the simplification, direct calculation gives

Consequently ${G^{\mathrm{gr}}}^{\pm }{D}^{\ast }=\mathcal{R}{{\mathsf{E}}_0}^{\pm }$ and DG^{\mathrm{gr}}^\pm D^*=\frac12\operatorname{id}. The last block cancels the algebraic contact term. The source elimination proves both Green identities; derivatives do not enlarge causal support.