Asymptotics of the triples equal-time correlation function for the turbulence developed in incompressible fluids in the region of widely separated wave vector values is investigated using the renormalization group approach and short-distance expansion. The problem of the most essential composite operators contributing to these asymptotics is examined. For this purpose, the critical dimensions of a family of composite quadratic tensor operators in the velocity gradient are found. Considered in the one-loop approximation, the contribution of these operators turns out to be less substantial (although not significantly) than the contribution of the linear term. The derived asymptotics of the triples correlator coincide in form with that predicted by the EDQNM approximation.