The simultaneous electron capture and the ejection of another electron by which a doubly charged He ion and a Ps atom are produced in a single collision between a positron and a He atom is investigated theoretically. The angular distributions of the Ps atom as well as of the ejected electron are studied in the low-and intermediate-͑100-500 eV͒ energy regime with respect to the threshold energy for the transfer ionization process. The electron-electron correlation effect which mainly governs such two-electron transition processes has been taken into account in both the initial and final channels. The fully differential cross sections for different dynamics reveal some structures which are to be verified by the future experiments.