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The quintet of Dirac 4 4 matrices suggests that the fundamental dimension of the internal (spin) space is n=5, instead of the conventional dimension n=4. Then extending the usual 4 4 tetrads (vierbein), gravity is described in terms of the 5-bein (f\"unfbein or five legs). We discuss the properties of the spacetime geometry induced from this 5-leg Dirac spinor theory, where the spin connection contains 10 4=40 elements instead of 24 elements in the tetrad formulation of the general relativity theory.
Obukhov et al. (Sun,) studied this question.