Tunneling time is an important factor to describe quantum electronic device. In this paper, the dynamic problem of spin-dependent tunneling is investigated by solving the time-dependent Schrödinger equation. The transmission coefficient and tunneling lifetime are discussed by use of mixing transfer-matrix and Runge-Kutta method. The k3 Dresselhaus term is considered to correct the effective Hamiltonian of the system in our calculation. The results show that the transmission peak of the electrons with different spin orientations split obviously. The building time and the tunneling lifetime through the double-barrier structure of AlxGa1-xSb material are different for the spin-down electron and spin-up electron. These time-dependent properties depend on the electronic spin orientation. It can be considered as one of reasons for spin polarization to appear. Additionally, the steady spin-polarization emerges in the well due to the k3 Dresselhaus spin-orbit coupling.