Maximal algebraic torus ($T$)

A maximal algebraic torus is an algebraic torus maximal among the tori contained in an algebraic group. In a connected reductive algebraic group, all maximal algebraic tori are conjugate. A Borel subgroup contains maximal algebraic tori, and its unipotent radical gives a decomposition $B=T⋉U$.

A maximal algebraic torus $T$ is rational for $F$ when $F(T)=T$. Rationality does not require every point of $T$ to be fixed: the finite group $T^F$ is its rational-point group. Similarly a rational Borel subgroup or parabolic subgroup means an $F$-stable algebraic subgroup.

A rational maximal torus is elliptic when it is contained in no proper rational parabolic subgroup. This definition allows the split central torus of the ambient group; an absolute absence of every split subtorus is not the right condition when the centre is nontrivial.