How does the Mystical Body of Christ relate to the 'People of God' in Lumen Gentium?

People of God is a more modern, general term for the church, used many times in Lumen Gentium, sometimes qualified as the people of God of the old covenant, the Israelites, or of the new covenant, the church. Mystici Corporis Christi was written in 1943, in the middle of World War II, when Pius XII was facing a turbulent time, and it reminded every Catholic that though we are separated by nations, languages, and cultures, and may never even meet, we are mystically one body, tied to Christ as the source, together with the communion of saints past and future. Vatican II, in Lumen Gentium in 1965, was not facing that kind of turbulence; it was trying to carry the gospel to the modern world, so it switched to the more inviting, general term people of God, and you will never see Vatican II calling the church the Mystical Body. The heart of people of God in Lumen Gentium is that they have charge of the gospel, the oracles of God, just as the Jews once did, a great privilege with great responsibility, because that custody has been transferred from the Jews to the church. So the terms overlap but suit different times: Mystical Body in 1943, people of God in 1965, and simply the church at other times, depending on context and audience.