  We give a representation theoretical proof of Branson's classification of
minimal elliptic sums of generalized gradients. The original proof uses tools
of harmonic analysis, which as powerful as they are, seem to be specific for
the structure groups SO(n) and Spin(n). The different approach we propose is
based on the relationship between ellipticity and optimal Kato constants and on
the representation theory of so(n). Optimal Kato constants for elliptic
operators were computed by Calderbank, Gauduchon and Herzlich. We extend their
method to all generalized gradients (not necessarily elliptic) and recover
Branson's result, up to one special case. The interest of this method is that
it is better suited to be applied for classifying elliptic sums of generalized
gradients of G-structures, for other subgroups G of the special orthogonal
group.
