  Let G be a semi-simple algebraic group over the complex numbers, B a Borel
subgroup of G, T a maximal torus in B and P a parabolic in G containing B. This
paper deals with singularities of T-stable subvarieties of G/P. It turns out
that under the restriction that G doesn't contain any G_2-factors, the key
geometric invariant determining the singular T-fixed points of X is the linear
span of the reduced tangent cone to X at a T-fixed point x provided the
singularity is isolated. The goal of this paper is to describe this invariant
at the maximal singular T-fixed points when X is a Schubert variety in G/P and
G doesn't contain any G_2-factors. We first describe the span of the tangent
cone solely in terms of Peterson translates, which were the main tool in a
previous paper. Then, taking a further look at the Peterson translates (with
the G_2-restriction), we are able to describe the span of the tangent cone at x
in terms of its isotropy submodule and the Bruhat graph of X at x. This
refinement gives a purely root theoretic description, which should be useful
for computations. It also leads to an algorithm for the singular locus of X.
