The purpose of the 'Starting up a Project' process is primarily to appoint people to the major roles within the project management team. This process ensures that the right people are in place with clear responsibilities before the project kicks off, facilitating effective planning and execution from the outset. References:
PRINCE2 7 Foundation manual and the 'Starting up a Project' process documentation.

A PRINCE2 project should consist of a start-up stage, an initiation stage, and at least one other management stage to cover the remainder of the project. This structure ensures that the project is properly initiated and planned before entering into a series of management stages where the work is executed and completed, followed by a closing stage. References: PRINCE2 Foundation manual, PRINCE2 official resources

The guiding obligations and good practices which determine whether a project is genuinely being managed using PRINCE2 form the 'Principles' of the method. PRINCE2 is founded on seven principles that must be present in the project management approach to consider the project being managed using PRINCE2.
References: PRINCE2 Foundation manual, PRINCE2 official resources

The "Directing a Project" process is used by the project board to respond to an exception report. This process involves the project board making decisions on how to proceed when project tolerances are forecasted to be exceeded, which are typically communicated through an exception report.
= PRINCE2 7 Foundation handbook and online resources

The Business Case in PRINCE2 documents the information needed to establish and maintain a project's continued business justification. It includes details on the expected benefits, costs, risks, and the rationale for the project, providing a basis to judge whether the project remains viable. References: PRINCE2 Foundation official guide and study material.