  Starting from the general concept of a Lie derivative of an arbitrary
differentiable map, we develop a systematic theory of Lie differentiation in
the framework of reductive G-structures P on a principal bundle Q. It is shown
that these structures 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 concept of a Lie
derivative of spinor fields is reviewed. On specializing to the case of the
Kosmann lift, we recover Kosmann's original definition. 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, and, as an interesting by-product, prove a result due to
Bourguignon and Gauduchon in a more general manner. Next, we give a new
characterization as well as a generalization of the Killing equation, and
propose a geometric reinterpretation of Penrose's Lie derivative of "spinor
fields". Finally, we present an important application of the theory of the Lie
derivative of spinor fields to the calculus of variations.
