Spacetime geometry from Dirac spinor theory

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