  We introduce the concept of strongly $r$-matrix induced ({\small SRMI})
Poisson structure, report on the relation of this property with the stabilizer
dimension of the considered quadratic Poisson tensor, and classify the Poisson
structures of the Dufour-Haraki classification (DHC) according to their
membership of the family of {\small SRMI} tensors. One of the main results of
our work is a generic cohomological procedure for {\small SRMI} Poisson
structures in arbitrary dimension. This approach allows decomposing Poisson
cohomology into, basically, a Koszul cohomology and a relative cohomology.
Moreover, we investigate this associated Koszul cohomology, highlight its tight
connections with Spectral Theory, and reduce the computation of this main
building block of Poisson cohomology to a problem of linear algebra. We apply
these upshots to two structures of the DHC and provide an exhaustive
description of their cohomology. We thus complete our list of data obtained in
previous works, see \cite{MP} and \cite{AMPN}, and gain fairly good insight
into the structure of Poisson cohomology.
