On page 79 of the document, at the beginning of the proof of Corollary 2.43, the authors claim that if $\overline{\rho}: G_{\mathbb Q} \to \operatorname{GL}_2(k)$ is a continuous absolutely irreducible representation, where $k$ is a finite field of characteristic $\ell$, then $H^0(G_{\mathbb Q}, \operatorname{ad}^0 \overline{\rho}(1)) = 0$ unless $\ell = 3$ and $\overline{\rho}|_{G_{\mathbb Q(\sqrt{-3})}}$ is not absolutely irreducible. Here, $\operatorname{ad}^0 \overline{\rho}$ is the subring of endomorphisms of the representation space of $\overline{\rho}$ given by trace zero endomorphisms, and the $(1)$ denotes a Tate twist.
Why is this true? I did sort of ask this before, but I did not get a complete answer (at least not that I understood).