Final answer:
The significant event that did NOT occur under British rule in the American colonies was the granting of the right to settle in the area west of the Allegheny and Appalachian Mountains. This denied the colonists access to the land they had fought for in the French and Indian War.
Step-by-step explanation:
One significant event that did NOT occur under British rule in the American colonies was the granting of the right to enter and settle in the area west of the Allegheny and Appalachian Mountains to the colonists. Despite their participation in the French and Indian War, the British government decided to tighten its control over the colonies and denied them access to the land they had fought for. This led to discontent and contributed to the growing tensions between Britain and the American colonies.