  We elaborate presentations of the interval groups related to all
quasi-Coxeter elements in the Coxeter group of type $D_n$. This is the only
case of the infinite families of finite Coxeter groups that admits proper
quasi-Coxeter elements. The presentations we obtain are over a set of
generators in bijection with what we call Carter generating set, and the
relations are those defined by the related Carter diagram along with a twisted
or a cycle commutator relator, depending on whether the quasi-Coxeter element
is a Coxeter element or not. In a subsequent work, we complete our analysis to
cover all the exceptional cases of finite Coxeter groups, and establish that
almost all the interval groups related to proper quasi-Coxeter elements are not
isomorphic to the related Artin groups, hence establishing a new family of
interval groups with nice presentations. Alongside the proof of the main
results of this paper, we establish important properties related to the dual
approach to Coxeter and Artin groups.
