The phase space of general relativity is first extended in a standard manner to incorporate spinors. New coordinates are then introduced on this enlarged phase space to simplify the structure of constraint equations. Now, the basic variables, satisfying the canonical Poisson-brackets relations, are the (density-valued) soldering forms σ~ a A B and certain spin-connection one-forms AaA B. Constraints of Einstein's theory simply state that σ~ a satisfies the Gauss law constraint with respect to Aa and that the curvature tensor FabA B and Aa satisfies certain purely algebraic conditions (involving σ~ a). In particular, the constraints are at worst quadratic in the new variables σ~ a and Aa. This is in striking contrast with the situation with traditional variables, where constraints contain nonpolynomial functions of the three-metric. Simplification occurs because Aa has information about both the three-metric and its conjugate momentum. In the four-dimensional space-time picture, Aa turns out to be a potential for the self-dual part of Weyl curvature. An important feature of the new form of constraints is that it provides a natural embedding of the constraint surface of the Einstein phase space into that of Yang-Mills phase space. This embedding provides new tools to analyze a number of issues in both classical and quantum gravity. Some illustrative applications are discussed. Finally, the (Poisson-bracket) algebra of new constraints is computed. The framework sets the stage for another approach to canonical quantum gravity, discussed in forthcoming papers also by Jacobson, Lee, Renteln, and Smolin.