[en] The formalism of differential geometry in the physical 4+4-dimensional superspace (SS) is developed on the basis of the simplest supergravity group of general transformations of the coordinates of the left and right chiral SS. In this formalism the only independent dynamical variable is the axial gravitational superfield (SF). All the geometrical objects - supertetrades, affinities, etc. - are expressed in terms of the axial gravitational SF. Analysis of the transformation properties of the spinor-coordinate derivatives has led to a natural definition of the local Lorentz group for SF. The simplest Lagrangian for pure supergravity is discussed