  We discuss the problem of the bias of the Internal Linear Combination (ILC)
CMB map and show that it is closely related to the coefficient of
cross-correlation K(l) of the true CMB and the foreground for each multipole l.
We present analysis of the cross-correlation for the WMAP ILC quadrupole and
octupole from the first (ILC(I)) and the third (ILC(III)) year data releases
and show that these correlations are about -0.52-0.6. Analysing 10^4 Monte
Carlo simulations of the random Gaussian CMB signals, we show that the
distribution function for the corresponding coefficient of the
cross-correlation has a polynomial shape P(K,l)\propto(1-K^2)^(l-1). We show
that the most probable value of the cross-correlation coefficient of the ILC
and foreground quadrupole has two extrema at K ~= +/-0.58$. Thus, the ILC(III)
quadrupole represents the most probable value of the coefficient K. We analyze
the problem of debiasing of the ILC CMB and pointed out that reconstruction of
the bias seems to be very problematic due to statistical uncertainties. In
addition, instability of the debiasing illuminates itself for the quadrupole
and octupole components through the flip-effect, when the even (l+m) modes can
be reconstructed with significant error. This error manifests itself as
opposite, in respect to the true sign of even low multipole modes, and leads to
significant changes of the coefficient of cross-correlation with the
foreground. We show that the CMB realizations, whose the sign of quadrupole
(2,0) component is negative (and the same, as for all the foregrounds), the
corresponding probability to get the positive sign after implementation of the
ILC method is about 40%.
