We are addressing the development of polarized lanthanum ($^{139}$La) target by applying a standard dynamic nuclear polarization (DNP) technique to a single crystal of lanthanum aluminate (LaAlO$_3$) doped with neodymium ions (Nd$^{3+}$), which is very attractive to searches of violating effects of time-reversal invariance using a slow neutron. For using the LaAlO$_3$ crystals as target material, it is very crucial to extend a spin-lattice relaxation time (T$_1$) through the optimization of the Nd$^{3+}$ concentration. We have almost established a method for precise control of the Nd$^{3+}$ amount and this method has allowed us to grow the crystal with 0.01 mol% concentration differently from the 0.05 mol% crystal. Simple DNP experiments by using a single-shot dewar at 1.3 K and 2.3 T have given the evaluation of the T$_1$ and the maximum polarization of these crystals, and then we have found that the 0.01 mol% concentration is the best one among the crystals tested in the past. Both of the T$_1$ and the maximum polarization in the 0.01 mol% crystal have not been exactly obtained because of the long relaxation time, so that the preparation of a new DNP system is in progress toward the DNP experiments for long-term measurement.