Should the E and B of executive branch be capitalized?

In TextCapitalize proper nouns (personal names, geographical names, organizations, etc.). However, many terms that are derived from or associated with proper nouns are not capitalized. See The Chicago Manual of Style for more detailed information.President A. B. Cee of XYZ Corporation

but A. B. Cee, president of XYZ Corporation

English standard units

but plaster of Paris

U.S. Department of Defense

but executive branch of the federal government

The official names of departments, projects, and programs should be capitalized appropriately.University and National Laboratory Programs department

Law department

Partnering for Total Quality program

Point-of-Use Ultrapure Water Monitoring project