Dynamics of amorphous cis-1,4-polybutadiene was studied by incoherent inelastic and quasielastic neutron scattering below 10 meV in a wide temperature range covering the glass transition temperature T(g). The spectra at low temperatures below T(g) show a broad excitation peak at about 2-3 meV, which is the so-called low-energy excitation and believed to be an origin of anomalous thermal properties of amorphous materials at low temperatures. It was found that the temperature dependence of the low-energy excitation intensity can be normalized by the Bose factor at temperatures low enough below T(g). With increasing temperature, a very fast motion appears in the energy range below approximately 4 meV at around the Vogel-Fulcher temperature T0, which is approximately 50 K below T(g). This fast motion is often interpreted as the beta process predicted by the mode coupling theory. In this paper, both the low-energy excitation and the fast motion were analyzed in terms of a common microscopic picture, i.e., an asymmetric double well potential. The natures of the two motions are discussed focusing on the relation between the two motions as well as the glass transition dynamics.