  We present the expressions for the monodromy matrix elements of the
six-vertex model in the F-basis for arbitrary Boltzmann weights. The results
rely solely on the property of unitarity and Yang-Baxter relations, avoiding
any specific parameterization of the weights. This allows us to write complete
algebraic expressions for the inner products and the underlying domain wall
partition functions in the case of arbitrary rapidities. We then apply our
results for the trigonometric six-vertex model in the presence of inhomogeneous
electric fields and obtain a determinant formula for the respective on-shell
scalar products.
