  Spinor formalism is the formalism induced by solutions of the Clifford
equation (the connecting operators). For the space-time manifold (n = 4), these
operators, connecting the tangent and spinor bundle, are operators that are
represented by the Dirac matrices in the special basis. Reduced connecting
operators are represented by the Pauli matrices. In order to uniquely prolong
the Killing equation from the tangent bundle onto the spinor bundle over the
space-time manifold, it is necessary to pass to the complexification of the
manifold and the corresponding bundles, and then to pass to the real
representation. Returning the reverse motion, one can already obtain two copies
of the spinor bundle. Their set (the pair-spinor) allows to construct the Lie
operator analogues for the spinors (and the pair-spinors). Similar procedure is
feasible for any even n. For n=6, the specified formalism is closely connected
with the Bogolyubov-Valatin transformations. For n mod 8=0, being based on the
Bott periodicity, the reduced connecting operators generate the structural
constants of an hypercomplex algebra (without division for n> 8) with the
alternative-elastic, flexible (Jordan), and "norm" identities. For n = 8, such
the algebra is the octonion algebra. In addition, in the article the various
options of the prolonging of the connection to the spinor bundle with
even-dimensional base are considered, and the corresponding curvature spinors
are constructed.
