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I don't have cardinals, but I have glolights with my lemon tetras and they school together only when they are nervous. All my tetras seem to loosely school.
Different tetras usually only school together because they have no other choice. They school for safety. They would much prefer to be with their own kind. I had a group of pristella tetras and 1 blackskirt. The blackskirt schooled with the pristellas until I added more blackskirts. Now occasionally they are together but not often.