The conventional two-component spinor approach to classical general relativity is utilized to construct a generalized torsion-free covarian-derivative structure. It is particularly shown that the structure can afford a geometric device which appropriately keeps track of arbitrary field quantities carrying both tensor and spinor indices. The device is then employed to express explicitly the curvature and Weyl spinors in terms of the densities that enter into the standard spin connections. An expression for the cosmological constant is also given.
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