Capitalize when it's a proper noun:
* If referring to a specific group with a defined name: For example, "The English Separatists who sailed on the Mayflower" would be capitalized because it's referring to a specific historical group.
* If it's used as a formal title: Similar to "the Democratic Party," "English Separatist" could be capitalized when used as a formal title or designation.
Don't capitalize when it's a general term:
* When describing the concept of separation from England: For instance, "Many people became English Separatists during the 17th century."
* When referring to a generic individual: "He was an English Separatist who believed in religious freedom."
In short:
* Capitalize when referring to a specific group or as a formal title.
* Don't capitalize when describing a concept or referring to individuals generally.
Always consider the context and use your best judgment.