  Motivated by situations with temporal evolution and spatial symmetries both
singled out, we develop a new 2+1+1 decomposition of spacetime, based on a
nonorthogonal double foliation. Time evolution proceeds along the leaves of the
spatial foliation. We identify the gravitational variables in the velocity
phase-space as the 2-metric (induced on the intersection $\Sigma_{t\chi }$ of
the hypersurfaces of the foliations), the 2+1 components of the spatial shift
vector, together with the extrinsic curvature, normal fundamental form and
normal fundamental scalar of $\Sigma _{t\chi }$, all constructed with the
normal to the temporal foliation. This work generalizes a previous
decomposition based on orthogonal foliations, a formalism lacking one metric
variable, now reintroduced. The new metric variable is related to (i) the angle
of a Lorentz-rotation between the nonorthogonal bases adapted to the
foliations, and (ii) to the vorticity of these basis vectors. As a first
application of the formalism, we work out the Hamiltonian dynamics of general
relativity in terms of the variables identified as canonical, generalizing
previous work. As a second application we present the unambiguous gauge-fixing
suitable to discuss the even sector scalar-type perturbations of spherically
symmetric and static spacetimes in generic scalar-tensor gravitational
theories, which has been obstructed in the formalism of orthogonal double
foliation.
