  We show that the Einstein-Hilbert, the Einstein-Palatini, and the Holst
actions can be derived from the Quadratic Spinor Lagrangian (QSL), when the
three classes of Dirac spinor fields, under Lounesto spinor field
classification, are considered. To each one of these classes, there corresponds
a unique kind of action for a covariant gravity theory. In other words, it is
shown to exist a one-to-one correspondence between the three classes of
non-equivalent solutions of the Dirac equation, and Einstein-Hilbert,
Einstein-Palatini, and Holst actions. Furthermore, it arises naturally, from
Lounesto spinor field classification, that any other class of spinor field
(Weyl, Majorana, flagpole, or flag-dipole spinor fields) yields a trivial
(zero) QSL, up to a boundary term. To investigate this boundary term we do not
impose any constraint on the Dirac spinor field, and consequently we obtain new
terms in the boundary component of the QSL. In the particular case of a
teleparallel connection, an axial torsion 1-form current density is obtained.
New terms are also obtained in the corresponding Hamiltonian formalism. We then
discuss how these new terms could shed new light on more general
investigations.
