We present an approach associated with the Jacobi matrix method to calculate a three-body wave function that describes the double continuum of an atomic two-electron system. In this approach, a symmetrized product of two Coulomb waves is used to describe the asymptotic wave function, while a smooth cutoff function is introduced to the dielectronic potential that enters its integral part in order to have a compact kernel of the corresponding Lippmann-Schwinger-type equation to be solved. As an application, the integral equation for the (${e}^{\ensuremath{-}},\phantom{\rule{4pt}{0ex}}{e}^{\ensuremath{-}},\phantom{\rule{4pt}{0ex}}{\mathrm{He}}^{2+}$) system is solved numerically; the fully fivefold differential cross sections (FDCSs) for ($e,3e$) processes in helium are presented within the first-order Born approximation. The calculation is performed for a coplanar geometry in which the incident electron is fast ($\ensuremath{\sim}6$ keV) and for a symmetric energy sharing between both slow ejected electrons at excess energy of 20 eV. The experimental and theoretical FDCSs agree satisfactorily both in shape and in magnitude. Full convergence in terms of the basis size is reached and presented.