Multiple vehicles control at an intersection is one of the most challenging scenarios in the vehicle control field due to the high number of constraints. However, regarding the control strategies of an isolated intersection, few studies have considered the longitudinal-lateral dynamics comprehensively. In the approaches that treat the turning curve as a straight line, a longitudinal dynamic model is adopted, while the lateral execution ability is ignored. The ideal decisions that ignore vehicle dynamics are not accurately executed by actual vehicles, as doing so would be futile. In addition, due to the strong coupling and nonlinearity of the longitudinal-lateral dynamics model, the optimization problem is difficult to solve. Therefore, in this paper, a reasonable model, the simplified vehicle dynamics model coupled with longitudinal-lateral dynamics is used to describe longitudinal, lateral, and yaw motions comprehensively. Optimization of the longitudinal and lateral control of each vehicle is realized simultaneously. The multi-objective optimization problem is decomposed and solved by each vehicle utilizing model predictive control (MPC) in a distributed manner. A novel safety constraint interpretive method (SCIM) is proposed to reduce the number of constraints and facilitate the solution by converting the safety constraint. The asymptotic stability of a local closed-loop system is guaranteed by a terminal constraint, and the global feasibility is proven. Finally, a demanding intersection scenario, with comparison between the Proportional-Integral-Derivative (PID) method and Adaptive Cruise Control (ACC) method, is carried out in a Processor-in-the-Loop (PiL) test. The results demonstrate that intersection safety is guaranteed with smoother control inputs obtained by the designed distributed method. The control inputs are more suitable for vehicle execution, especially in turning maneuvers.