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This is untrue in the case of Williamson. He'd only been an MP since 2010, and only aligned himself with the left some time after that. Prior to his career as an MP he'd been on Derby council, and was notably an outspoken advocate of PFI.
In the case of Livingstone, yes, he'd been historically more aligned with Corbyn's politics, but let's not pretend he sprung up from nowhere. He was Mayor until 2004...
There is some failing to see the wood from the trees here. The likes of Livingstone and Williamson were lifelong political allies and fellow travelers of Corbyn. The crisis with anti-semitism in the party arose under Corbyn's leadership because of the people he gave prominence too and the people he attracted to the party.