This might be about Jane Seymour, the third wife of King Henry VIII. She was one of the few wives not executed, but died at childbirth while bearing his only son. Another possibility is that it's about Jane Ormsby-Gore, a British woman Mick Jagger was involved with.
This song was rumored to be about drugs, with "Lady Anne" code for amphetamine and "Lady Jane" a reference to marijuana (as in "Mary Jane").
Brian Jones, who was The Stones guitarist until his death in 1969, played the dulcimer, an instrument you play on your lap by plucking or strumming the strings. Jones could learn just about any instrument very quickly. He had just recently learned how to play it when they recorded this.
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