The teleparallel coframe gravity may be viewed as a generalization of the standard
GR. A coframe (a field of four independent 1-forms) is considered, in this approach,
to be a basic dynamical variable. The metric tensor is treated as a secondary
structure. The general Lagrangian, quadratic in the first order derivatives of the
coframe field is not unique. It involves three dimensionless free parameters. We
consider a weak field approximation of the general coframe teleparallel model. In
the linear approximation, the field variable, the coframe, is covariantly reduced to
the superposition of the symmetric and antisymmetric field. We require this reduction
to be preserved on the levels of the Lagrangian, of the field equations, and of
the conserved currents. This occurs if and only if the pure Yang–Mills-type term is
removed from the Lagrangian. The absence of this term is known to be necessary
and sufficient for the existence of the viable (Schwarzschild) spherical-symmetric
solution. Moreover, the same condition guarantees the absence of ghosts and tachyons
in particle content of the theory. The condition above is shown recently to
be necessary for a well-defined Hamiltonian formulation of the model. Here we
derive the same condition in the Lagrangian formulation by means of the weak field
reduction.