  Exotic quantum phases including topological states and non-Fermi liquids may
be realized by quantum states with total angular momentum $j=3/2$, as
manifested in HgTe and pyrochlore iridates. Recently, an exotic superconducting
state with non-zero density of states of zero energy Bogoliubov (BG)
quasiparticles, Bogoliubov Fermi-surface (BG-FS), was also proposed in a
centrosymmetric $j=3/2$ system, protected by a Z$_2$ topological invariant.
Here, we consider interaction effects of a centrosymmetric BG-FS and
demonstrate its instability by using mean-field and renormalization group
analysis. The Bardeen-Cooper-Schrieffer (BCS) type logarithmical enhancement is
shown in fluctuation channels associated with inversion symmetry. Thus, we
claim that the inversion symmetry instability is an intrinsic characteristic of
a BG-FS under generic attractive interactions between BG quasiparticles. In
drastic contrast to the standard BCS superconductivity, a Fermi-surface may
generically survive even with the instability. We propose the experimental
setup, a second harmonic generation experiment with a strain gradient, to
detect the instability. Possible applications to iron based superconductors and
heavy fermion systems including FeSe are also discussed.
