We assume that the conditions of questions 12 and 14 are satisfied and that $\lambda(0) \neq 0$. Let $C = (q - q^{-1}) E \circ F + q^{-1} H + q H^{-1}$ with $H^{-1}$ the inverse of $H$.
16a. Show that $C = (q - q^{-1}) F \circ E + q H + q^{-1} H^{-1}$.
16b. For $i \in \mathbf{Z}$, show that $v_i$ is an eigenvector of $C$.
16c. Deduce that $C$ is a homothety of $V$ and calculate its ratio $R(\lambda(0), \mu(0), q)$ in terms of $\lambda(0), \mu(0)$ and $q$.
16d. We fix $q$ and $\lambda(0)$. Show that the map $\mu(0) \mapsto R(\lambda(0), \mu(0), q)$ is a bijection from $\mathbf{C}$ to $\mathbf{C}$.
16e. We fix $q$ and $\mu(0)$. Show that the map $\lambda(0) \mapsto R(\lambda(0), \mu(0), q)$ is a surjection from $\mathbf{C}^*$ to $\mathbf{C}$ but not a bijection.