  We consider a bi-connection model in the presence of four-dimensional
Gauss-Bonnet term adding to the Einstein-Hilbert action. This generalization
solves the dynamics issue which exists in pure Einstein-Hilbert formalism of
bi-connection model. As an example we study the Weyl inspired bi-connection
model and show there is a self-accelerating solution in this model. To compare
it with previous results we try to find appropriate generalization of the Weyl
geometrical bi-connection model to reach at de Rham-Gabadadze-Tolley massive
gravity. In this formalism mixing terms between the potential and kinetic terms
appear automatically.
