  We consider the non-Abelian Bianchi identities in SU(2) pure Yang-Mills
theory in D=3,4 focusing on the possibility of their violation and the
significance of the chromomagnetic fields degeneracy points. We show that the
recently proposed non-Abelian Stokes theorem allows to formulate the Bianchi
identities in terms of the physical fluxes and their relative color
orientations. Then the violation of Bianchi identities becomes a well defined
concept ultimately related to the degeneracy points. The locality and gauge
invariance of our approach allows to study the problem numerically. We present
evidences that in D=4 the suppression of the Bianchi identities violation is
likely to destroy confinement while the removal of the degeneracy points drives
the theory to the topologically non-trivial sector. However, confronting the
results obtained in three and four dimensions we argue that it is the mass
dimension two condensate <A^2_{min}> which probably explains our findings.
