  The names tetrad, tetrads, cotetrads, have been used with many different
meanings in the physical literature, not all of them, equivalent from the
mathematical point of view. In this paper we introduce unambiguous definitions
for each one of those terms, and show how the old miscellanea made many authors
to introduce in their formalism an ambiguous statement called `tetrad
postulate', which has been source of many misunderstandings, as we show
explicitly examining examples found in the literature. Since formulating
Einstein's field equations intrinsically in terms of cotetrad fields theta^{a},
a = 0,1,2,3 is an worth enterprise, we derive the equation of motion of each
theta^{a} using modern mathematical tools (the Clifford bundle formalism and
the theory of the square of the Dirac operator). Indeed, we identify (giving
all details and theorems) from the square of the Dirac operator some noticeable
mathematical objects, namely, the Ricci, Einstein, covariant D'Alembertian and
the Hodge Laplacian operators, which permit to show that each theta^{a}
satisfies a well defined wave equation. Also, we present for completeness a
detailed derivation of the cotetrad wave equations from a variational
principal. We compare the cotetrad wave equation satisfied by each theta^{a}
with some others appearing in the literature, and which are unfortunately in
error.
