Electron-impact direct double ionization (DDI) process is studied as a sequence of two and three step processes. Contribution from ionization-ionization, ionization-excitation-ionization, and excitation-ionization-ionization processes is taken into a
ccount. The present results help to resolve the long-standing discrepancies; in particular, a good agreement with experimental measurements is obtained for double ionization cross-sections of $O^{1+}$, $O^{2+}$, $O^{3+}$, $C^{1+}$, and $Ar^{2+}$ ions. We show that distribution of the energy of scattered and ejected electrons, which participate in the next step of ionization, strongly affects DDI cross-sections.
