+ (\eta I_4 + \tau \beta) \mathcal{C}_\lambda\big]
      &\supset \operatorname{ran}\big[(I_4 + (\eta I_4 + \tau \beta) \mathcal{C}_\lambda)
      (I_4 - (\eta I_4 + \tau \beta) \mathcal{C}_\lambda) \big] \\
      &= \operatorname{ran}\big[I_4 - \big( (\eta I_4 + \tau \beta) \mathcal{C}_\lambda \big)^2 \big]
    \end{split}$$ holds. Note that $I_4 - \big( (\eta I_4 + \tau \beta) \mathcal{C}_\lambda \big)^2$ is injective, as otherwise $\lambda$ would be a non-real eigenvalue of one of the symmetric operators $A_{\eta, \tau}$ or $A_{-\eta, -\tau}$ by (ii). Moreover, $$\begin{split}
      I_4 - \big( (\eta I_4 + \tau \beta) \mathcal{C}_\lambda \big)^2
        &= I_4 - \tau (\mathcal{C}_\lambda \beta + \beta \mathcal{C}_\lambda) (\eta I_4 + \tau \beta) \mathcal{C}_\lambda - (\eta^2 - \tau^2) \mathcal{C}_\lambda^2 \\
      &= \left(1 - \frac{\eta^2 - \tau^2}{4 c^2} \right) I_4 + \mathcal{K}_\lambda,
    \end{split}$$ where $\mathcal{K}_\lambda$ is a compact operator in $L^2(\Sigma)^4$ by Proposition \[proposition\_mapping\_properties\]. Therefore, Fredholm’s alternative implies that $I_4 - \big( (\eta I_4 + \tau \beta) \mathcal{C}_\lambda \big)^2$ is also surjective. From  we deduce that the injective operator $I_4 + (\eta I_4 + \tau \beta) \mathcal{C}_\lambda$ is also surjective, which yields the claim of assertion (iii).

Now we are prepared to show the self-adjointness of $A_{\eta, \tau}$ in the case of noncritical interaction strengths. Moreover, we prove an explicit Krein type resolvent 