  In the double field theory, gauge symmetries are realized as generalized
diffeomorphisms in the doubled spacetime. By consistency of the theory,
dependence of tensor fields on the doubled coordinates is strongly constrained.
This causes finite transformation law highly complicated, both technically and
conceptually. In this paper, we propose a new, physically intuitive approach to
finite gauge transformations by utilizing untwisted form of vector fields. In
our approach, finite gauge transformation law is expressed in terms of
diffeomorphisms on a maximal null subspace on which dynamical degrees of
freedom live and of a local rotation of this null subspace embedded inside the
doubled space. We show that our finite transformation automatically satisfies
the composition law. We also show that ours is free from the so-called
Papadopoulos problem, so can describe background with nontrivial three-form
flux $H_3$. Added advantage of our approach is straightforward applicability to
general extended field theories. We demonstrate this by explicitly obtaining
finite transformation law in the SL(5) exceptional field theory.
