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

Strongly r-matrix induced tensors, Koszul cohomology, and arbitrary-dimensional quadratic Poisson cohomology