Ms. Abedin, a longtime Clinton confidante, served as Secretary Clinton’s deputy chief-of-staff at the State Department and is now vice chairwoman of Mrs. Clinton’s 2016 campaign for the presidency. For about a half-year toward the end of Abedin’s tenure at the State Department, which coincided with Secretary Clinton’s own departure from the State Department in February 2013, Abedin was given a highly unusual and ethically dubious arrangement. She was permitted to work simultaneously for the State Department, Teneo, the Clinton Foundation, and even in a personal capacity for Mrs. Clinton in order to manage the secretary’s transition back to the...