$\lambda$-quiddity and subgroups generated by an algebraic number

During his work devoted to Coxeter's friezes, M. Cuntz initiated the study of the notion of $\lambda$-quiddity and raised the problem of the study of this over some subsets of $\mathbb C$. More specifically, $\lambda$-quiddities are the solutions to a matrix equation, related to various mathematical objects, which we seek to solve over different sets. The aim of this text is to provide some new insights into the problem raised by M. Cuntz in the case of some cyclic subgroups of ($\mathbb{C},+$) generated by an algebraic number. In particular, we will study the cases of subgroups generated by $a+b\sqrt{k}$.