In the context of algebraic geometry, decomposition and inertia groups are special subgroups of the Cremona group which preserve a certain subvariety of Pn as a set and pointwise, respectively. These groups were and still are classic objects of study in the area, with explicit descriptions in several instances. In the particular case where this fixed subvariety is a hypersurface of degree n+1, we have the notion of Calabi-Yau pair which allows us to use new tools to deal with the study of these groups and one of them is the so-called volume preserving Sarkisov Program. Using this approach we prove that an appropriate algorithm of the Sarkisov Program in dimension 2 applied to an element of the decomposition group of a nonsingular plane cubic is automatically volume preserving. From this, we deduce some properties of the (volume preserving) Sarkisov factorization of its elements. Regarding now a 3-dimensional context, we give a description of which toric weighted blowups of a point are volume preserving and among them, which ones will initiate a volume preserving Sarkisov link from a Calabi-Yau pair (P3,D) of coregularity 2. In this case, D is necessarily an irreducible normal quartic surface having canonical singularities. This last result enhances and extends the recent works of Guerreiro and Araujo, Corti and Massarenti in a log Calabi-Yau geometrical perspective, and it is a possible starting point to study the decomposition group of such quartics.