Abstract This is the first of three companion papers describing a complete and modular library to carry out the four-index transformation of a block [( pq/mrs )] of two-electron integrals involving symmetry-adapted primitive orbitals p, q, r, s and a symmetry-related index m into a block [( ij/klm )] of integrals over symmetry-adapted orthonormal (orbitals i, j, k, l . The index m allows for the unified treatment of atomic and molecular, relativistic and nonrelativistic calculations. A new algorithm based on a generalization of a method by Saunders & van Lenthe [ Mol. Phys. 48, 923 (1983)] for non-symmetry orbitals has been developed. It is here implemented by means of a FORTRAN code, controlled by subroutine C4ITD, requiring all computed quantities to be held in central memory. A non-redundant set of integrals is used throughout. All pertinent sums run only over a limited range determined by a given symmetry. The rate determining steps may be carried out in parallel and are vectorizable within each possible concurrent processor. For 20 non-symmetry orbitals and with 64-bit precision on a VAX-11/780 computer, using a working set of 200 kbytes, subroutine C4ITD takes 48 CPU s. Ninety percent of this time is spent in steps dominated by vector-scalar-multiply-and-add (VSMA) operations carried out at 0.170 mega floating point operations per second, which is 91% of the performance achieved on the same computer by an equivalent mix of pure VSMA operations and vector sizes. C4ITD will be useful in atomic and molecular electronic structure calculations beyond Hartree-Fock.