  We study the Internal Linear Combination (ILC) method presented by the
Wilkinson Microwave Anisotropy Probe (WMAP) science team, with the goal of
determining whether it may be used for cosmological purposes, as a
template-free alternative to existing foreground correction methods. We
conclude that the method does have the potential to do just that, but great
care must be taken both in implementation, and in a detailed understanding of
limitations caused by residual foregrounds which can still affect cosmological
results. As a first step we demonstrate how to compute the ILC weights both
accurately and efficiently by means of Lagrange multipliers, and apply this
method to the observed data to produce a new version of the ILC map. Next we
describe how to generate Monte Carlo simulations of the ILC map, and find that
these agree well with the observed map on angular scales up to l~200. Finally
we make two comments to the on-going debates concerning the large-scale
properties of the WMAP data. First, we note that the Galactic south-eastern
quadrant is associated with notably different ILC weights than the other three
quadrants, possibly indicating a foreground related anisotropy. Second, we
study the properties of the quadrupole and octopole (amplitude, alignment and
planarity), and confirm the previously reported anomalies. Even more
interestingly, we find that the l=5 mode is spherically symmetric at about 3
sigma, and that the l=6 mode is planar at the 2 sigma level. However, we also
assess the impact of residual foregrounds on these statistics, and find that
the ILC map is not clean enough to allow for cosmological conclusions.
Alternative methods must be developed to study these issues further.
