Let $C$ be an invertible integer matrix. Then a matrix $M$ is called Coxeter matrix (following Sato in https://www.sciencedirect.com/science/article/pii/S0024379505001709?via%3Dihub ) when $M=-C^{-1} C^T$ for such a $C$.
Then a Coxeter matrix $M$ is weakly periodic when $M^k-id$ is nilpotent for some $k \geq 1$. This is equivalent to the characteristic polynomial of $M$ being a product of cyclotomic polynomials and equivalent to that all eigenvalues of $M$ have absolute value 1.
Question: Is there a similar characterisation when all non-real eigenvalues of a Coxeter matrix $M$ have absolute value 1?
I noted that this happens rather frequently for Coxeter matrices of finite dimensional algebras that all non-real eigenvalues have absolute value 1 (but most real eigenvalues do not have this property).
