On the spinor formalism for even n
  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. 
K. V. Andreev
math-ph math.DG math.MP