In acetylacetone the energy level matching spectrum of nuclear Zeeman and tunneling energies of CH3 groups in the rotating frame of their proton spins demonstrates resonant transfers of population characteristic of a 100 KHz range tunneling splitting. However, in this material two A to E tunneling splittings, of 10.4 GHz and 1 GHz, were observed by inelastic neutron scattering. Therefore the small energy splitting must be a result of a weak torsion-torsion interaction, which has not been resolved in neutron spectroscopy. This small splitting is investigated through the dependencies of its energy level matching spectra on mixing time, tilt angle in the magnetic field, temperature, and the strength of the main de field H-0. Upon heating, the level matching resonances begin to broaden at 10 K and average out to zero at 25 K. The spectrum does not depend on H-0. These observations are explained by Zeeman-tunneling level matching transitions within manifolds of EEE-symmetry states and AE-symmetry states, both of which are split by the interaction among three CH3 groups. Based on resonances detected at nu(z) = n Delta, where n = 1/4, 1/3, 1/2, and 1, this splitting Delta is determined to be 116 +/- 2 KHz at 10 K.