Is there more to Dirac's gamma matrices than meets the eye? It turns out that the gamma zero operator can be split into three components. This revelation facilitates the expansion of Dirac's space-time algebra to Clifford algebra Cl(0,6). The resultant rich geometric structure can be leveraged to establish a combined framework of gravity and beyond the standard model, wherein a gravi-weak interaction between the vierbein field and the extended weak gauge field is allowed. Inspired by the composite Higgs model, we examine the vierbein field as an effective description of the fermion-antifermion condensation. The compositeness of space-time manifests itself at an energy scale which is different from the Planck scale. We propose that the regular Lagrangian terms including the cosmological constant are of quantum condensation origin, thus possibly addressing the naturalness problem. The Clifford algebra approach also permits a weaker form of charge conjugation without particle-antiparticle interchange, which leads to a Majorana-type mass that conserves lepton number. Additionally, in the context of spontaneous breaking of two global U(1) symmetries, we explore a three-Higgs-doublet model which could explain the fermion mass hierarchies.