  We examine nucleon-nucleon realistic interactions, based on their SU(3)
decomposition to SU(3)-symmetric components. We find that many of these
interaction components are negligible, which, in turn, allows us to identify a
subset of physically relevant components that are sufficient to describe the
structure of low-lying states in $^{12}$C and related observables, such as
excitation energies, electric quadrupole transitions and rms radii. We find
that paring the interaction down to half of the SU(3)-symmetric components or
more yields results that practically coincide with the corresponding ab initio
calculations with the full interaction. In addition, we show that while various
realistic interactions differ in their SU(3) decomposition, their renormalized
effective counterparts exhibit a striking similarity and composition that can
be linked to dominant nuclear features such as deformation, pairing,
clustering, and spin-orbit effect.
