That particular article is mining English translations for meanings not implied in the original -- the first verse referenced, for example, is not talking about being brought to life after having lived before, but instead is quite plainly referencing just the most basic creation narrative: life from non-life. There is no "back from a prior state of life" about it, so to be honest I only skimmed from there on after because once you assert that the Qur'an absolutely demands belief in something based on a sort of shady translation ... well. Â That's not really going to go over well with anyone.
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It's possible reincarnation has worked its way into certain sects and offshoots, particularly where there is more cultural overlap with other belief systems which adhere to the idea, but it's not something inherent in Islam itself.