The white leftists don't identify as white because, again, it's not actually about skin color. They call themselves allies or antiracist. It's the same way that conservative blacks are not considered black. You are still trying to process this through the lens of race and it will never make sense that way. "White" is a group that has benefitted from capitalism and doesn't want to change the system because it's the source of their political power. Leftists who are Caucasian absolve themselves of that sin by allying with other minority groups and calling for equity via some form of globalist socialism.

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