Complex sentence: After the war, LADA became part of a multi-department defensive system, run by a War Office department known as Air Defence of Great Britain (ADGB), which was responsible for the defence of the British Isles during the 1920s and 30s.
Split into simpler sentences: After the war, LADA was run by a War Office department known as Air Defence of Great Britain (ADGB). ADGB was responsible for the defence of the entire British Isles during the 1920s and 30s.