Cartan subgroup (awaiting review)

A Cartan subgroup is a maximal abelian subgroup that is semisimple (diagonalizable over the algebraic closure). There are two types of Cartan subgroups of $\GL(2,\F_p)$, split Cartan subgroups, which are diagonalizable over $\F_p$, and non-split Cartan subgroups, which are diagonalizable over $\F_{p^2}$.