Complex sentence: The day to day running of the RDA was the responsibility of the Chief Executive who was appointed by the Board, subject to approval by BIS ministers (or the London Mayor in the case of the LDA).
Split into simpler sentences: The day to day running of the RDA was the responsibility of the Chief Executive. The Chief Executive was appointed by the Board, subject to approval by BIS ministers (or the London Mayor in the case of the LDA).