A Cartan is not “any maximal connected abelian subgroup.” That works in a compact group, where every abelian subgroup is a subgroup of a Cartan. In a linear algebraic group there are not only diagonalizable elements, but also unipotent elements and subgroups of them. Cartans in algebraic groups are not contained in larger abelian subgroups, but there are unipotent subgroups that are not contained in them and which have much larger dimension. I think people usually define them as maximal tori, that is, as (abelian subgroups) consisting only of diagonalizable elements, but Cartan had a different definition involving self-normalization.
typos: the natural representation of A_n is C^n+1, not C^n; in the last line outside a remark, a “times” should be a “\times”

*[Ah, that was a subtlety I was not aware of! Thanks for the corrections – T.]*

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