When whites mockingly use the phrase "so woke" to Blacks, that illustrates the core of white supremacy for whites of every ethnicity and every nation of origin.
Hence, Black sociologists such as Dr. Eduardo Bonilla-Silva (author of Racism without Racists) have written about this for a century to remove the ability of whites of any ethnicity and of any nation of origin to declare themselves immune.
Whites oppressing each other (British harming Irish) does not remove the ability for ethnic groups to identify as white and to unite for racial power across most contexts and across most institutions around the world (which is the meaning of racism--stop using white dictionary definitions).
Even Dr. Noel Ignatiev's How the Irish became White provides one perspective of Irish assimilation into whiteness based on the promise of being superior to Blacks and being able to be in control.
This basic understanding would be more prevalent if most history books and most history classes were not white supremacist.