In this paper, using E. Carten's exterior calculus, we give the spinor form of the structure equations, which leads naturally to the Newman-Penrose equations. Further, starting from the spinor space and the sl(2C) algebra, we construct the general complex-vector formalism of general relativity. We find that both the Cahen-Debever-Defrise complex-vector formalism and the Brans one are its special cases. Thus, the spinor formalism and the complex-vector formalism of general relativity are unified on the basis of the unimodular group SL(2C) and its Lie algebra.
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