Faithfulness criterion for a block representation of the cyclic group of order four ($T(a)=\mathrm{diag}\left(1,\left(\begin{array}{cc}0& b\\ c& 0\end{array}\right)\right)$)

For real $b,c$, this matrix defines a group representation of $C_4=⟨a:a^4=e⟩$ exactly when $bc=\pm 1$. Indeed its lower block $B$ satisfies $B^2=bc{I}_2$, so $T(a{)}^4=I_3$ exactly when $(bc{)}^2=1$. If $bc=-1$, $T(a{)}^2=\mathrm{diag}(1,-1,-1)\ne I_3$, so its order is four and the representation is faithful. If $bc=1$, its order is exactly two and the kernel of a group homomorphism is $\{e,a^2\}$, giving an unfaithful representation. Other values do not give representations at all, including $bc=0$, when the proposed matrix is singular.