  Reductive G-structures on a principal bundle Q are considered. It is shown
that these structures, i.e. reductive G-subbundles P of Q, admit a canonical
decomposition of the pull-back vector bundle $i_P^*(TQ) = P \times_Q TQ$ over
P. For classical G-structures, i.e. reductive G-subbundles of the linear frame
bundle, such a decomposition defines an infinitesimal canonical lift. This lift
extends to a prolongation $\Gamma$-structure on P. In this general geometric
framework the theory of Lie derivatives is considered. Particular emphasis is
given to the morphisms which must be taken in order to state what kind of Lie
derivative has to be chosen. On specializing the general theory of
gauge-natural Lie derivatives of spinor fields to the case of the Kosmann lift,
we recover the result originally found by Kosmann. We also show that in the
case of a reductive G-structure one can introduce a "reductive Lie derivative"
with respect to a certain class of generalized infinitesimal automorphisms.
This differs, in general, from the gauge-natural one, and we conclude by
showing that the "metric Lie derivative" introduced by Bourguignon and
Gauduchon is in fact a particular kind of reductive rather than gauge-natural
Lie derivative.
