Complex sentence: Spain, which ruled much of Italy at the time, raised the claim in 1605 and again in 1644 at the conclave that elected Cardinal Giovanni Battista Pamphili (who became Pope Innocent X) to exclude Cardinal Sacchetti.
Split into simpler sentences: Spain, which ruled much of Italy at the time, raised the claim in 1605. In 1644, in the conclave which elected Cardinal Giovanni Battista Pamphili (who became Pope Innocent X), "jus exclusivae" was first exercised, by Spain, to exclude Cardinal Sacchetti.