 Maximal algebraic torus

A maximal algebraic torus is an algebraic torus not properly contained in another algebraic torus of the ambient algebraic group. Over an algebraically closed field, maximal algebraic tori in a connected reductive algebraic group are conjugate. A Borel subgroup contains one, and it splits as $B=T⋉R_u(B)$ with its unipotent radical.

For $G{L}_n$, the invertible diagonal matrices form a maximal algebraic torus isomorphic to ${\mathbb{G}}_m^n$. Choosing a different basis conjugates this example. The dimension of a maximal algebraic torus is the absolute rank of the connected reductive group.

Over a general field, geometric conjugacy does not assert conjugacy by rational points. In $S{L}_2(\mathbb{R})$, the split diagonal algebraic torus has noncompact real points, while the rotation norm-one algebraic torus has compact real points, so they cannot be conjugate by a real matrix. Both become maximal split algebraic tori after passing to the complex numbers.^[ref]