Abstract: We explore geometric realizations of supersymmetry in the adjoint representation, beginning with a review of the fundamentals of this approach. We then consider a unified formulation of gravity, Yang–Mills fields, and fermionic matter based on an osp(N|4)-valued superconnection. A Clifford lift of the algebra allows us to construct the action as a sum of Clifford-generated invariants, linking the structure and relative coefficients of the gravity, gauge, and matter sectors. Suitable parameter choices reproduce known quadratic gravity models with a ghost-free linearized spectrum in the reduced bosonic sector, while eliminating the independent second-order fermionic kinetic term. This construction provides a geometric route to unification without extra dimensions or an externally introduced symmetry-breaking field.

A central ingredient is the Clifford lift e^a\gamma_a of the coframe. Its Lorentz equivariance identifies the vector transformation of the coframe with conjugation in the spinor representation and extends naturally to its exterior powers. We then use this structure to develop chiral gravity based on osp(1|2)_L \oplus osp(1|2)_R. The exterior algebra of the Clifford-valued coframe generates the relevant Lorentz modules and invariant differential forms, including the self-dual and anti-self-dual Plebański two-forms. Their simplicity relations hold identically, so no separate simplicity constraints need to be imposed. Appropriate combinations of the chiral invariants reconstruct Einstein–Cartan gravity with a cosmological constant and the Holst term.

Finally, we discuss the matter ansätz for spinorial one- and two-forms that express the fermionic sector in terms of spin-1/2 fields. In the chiral construction, a spinorial BF-type pairing supplies their first-order kinetic term and torsion coupling. We examine the residual supersymmetry transformations compatible with these ansätze, emphasizing how preserving the one-form ansatz while holding the coframe fixed leads to the twistor-spinor equation.