From the Lorentz invariant to the coframe form of $f(T)$ gravity

It is shown that the Lorentz invariant $f(T)$ gravity, defined by the coframe-connection-multiplier form of the Lagrangian, can be gauge-fixed to the pure coframe form. After clarifying basic aspects of the problem in the Lagrangian formalism, a more detailed analysis of this gauge equivalence is given relying on the Dirac Hamiltonian approach.