  Spinor structure is understood as a totality of tensor products of
biquaternion algebras, and the each tensor product is associated with an
irreducible representation of the Lorentz group. A so-defined algebraic
structure allows one to apply modulo 8 periodicity of Clifford algebras on the
system of real and quaternionic representations of the Lorentz group. It is
shown that modulo 8 periodic action of the Brauer-Wall group generates modulo 2
periodic relations on the system of representations, and all the totality of
representations under this action forms a self-similar fractal structure. Some
relations between spinors, twistors and qubits are discussed in the context of
quantum information and decoherence theory.
