Dirichlet's unit theorem computes the group of units of the algebraic numbers of a number field. There are a few generalisations for orders available.

Assume the order has an involution. For example, the number field $\mathbb{Q}(\xi_1,\dots,\xi_n)$ can be equipped with an involution $\xi_k \mapsto \overline{\xi_k} = \pm \xi_k$ and the ring of algebraic numbers will (hopefully) inherit this involution.

It is natural to ask for the "unitary units" of the order i.e. all $x$ such that $x\overline{x}=1$.

Is there a theorem that tells us the structure of the group of unitary units? Do we know at least when this group is (in-)finite?