  Coupling spinor fields to the gravitational field, in the setting of general
relativity, is standardly done via the introduction of a vierbein field and the
(associated minimal) spin connection field. This makes three types of indices
feature in the formalism: world/coordinate indices, Lorentz vector indices, and
Lorentz spinor indices, respectively. This article will show, though, that it
is possible to dispense altogther with the Lorentz indices, both tensorial ones
and spinorial ones, obtaining a formalism featuring only world indices. This
will be possible by having both the 'Dirac operator' and the generators of
'Lorentz' transformations become spacetime-dependent, although covariantly
constant. The formalism is developed in the setting of complexified
quaternions.
