The conclusion is that every element of the complexified C l 3 , 1 ( R ) in End(H) (i.e. every complex 4 × 4 matrix) has well defined Lorentz transformation properties. In addition, it has a spin-representation of the Lorentz Lie algebra, which upon exponentiation becomes a spin representation of the group, acting on C 4 , making it a space of bispinors.