A standard Cholesky decomposition of the two-electron integral matrix leads to integral tables which have a huge number of very small elements. By neglecting these small elements, it is demonstrated that the recursive part of the Cholesky algorithm is no longer a bottleneck in the procedure. It is shown that a very efficient algorithm can be constructed when family type basis sets are adopted. For subsequent calculations, it is argued that two-electron integrals represented by Cholesky integral tables have the same potential for simplifications as density fitting. Compared to density fitting, a Cholesky decomposition of the two-electron matrix is not subjected to the problem of defining an auxiliary basis for obtaining a fixed accuracy in a calculation since the accuracy simply derives from the choice of a threshold for the decomposition procedure. A particularly robust algorithm for solving the restricted Hartree-Fock (RHF) equations can be speeded up if one has access to an ordered set of integral tables. In a test calculation on a linear chain of beryllium atoms, the advocated RHF algorithm nicely converged, but where the standard direct inversion in iterative space method converged very slowly to an excited state.