We have developed a theory model for a three-element laser array where three lasers are laterally coupled using the coupled mode theory and Maxwell equations. New chaotic synchronization properties have been observed systematically in the master-slave configuration, consisting of the driving three-element laser array with self-feedback and the response three-element laser array subjected to the parallel injection or cross injection. Under the parallel injection, the dynamic evolutions of high-quality complete chaotic synchronization between laser elements in different parameter spaces seriously depend on the self-feedback mode of the driving laser elements, such as one, two and all of them with self-feedback. It is found that when only the driving middle one or all of the driving laser elements are subject to self-feedback, high-quality complete chaotic synchronization of all laser elements can be achieved in the same large region of the most of the parameter spaces. In addition, we report here for the first time (to our knowledge) the interestingly symmetrical properties of leader/ laggard chaotic synchronization in the configuration under the cross-injection. Namely, the leader/ laggard chaotic synchronization with high quality between laser elements periodically varies with the delay differences, under the key parameters limited to a certain range. The varying traces of these synchronizations are like sine wave. The mirror symmetry between the laggard chaotic synchronization with in-phase (anti-phase) and the leader one with in-phase (anti-phase) can be achieved by the optimization of the structural parameters of laser waveguides. With the optimization of the related operating parameters, for one of the side-lasers, its leader/ laggard chaotic synchronization can be achieved the anti-symmetry between in-phase and anti-phase. On the other hand, for two symmetrical side-lasers, their leader/ laggard chaotic synchronization with in-phase and anti-phase can reach the anti-symmetry.