A pretty naive question: An anisotropic algebraic (linear) group over a field $k$ is a linear algebraic group $G$ defined over $k$ and of $k$-rank zero, i.e. not containing non-trivial $k$-split tori (Ref.: A. Borel, Linear algebraic groups, Zbl 0186.33201).
Question: What is the intuition behind this terminology in context of algebraic groups? Etymologically isotropic is Greek for "the same in every direction."
In other fields of mathematics the same name is used also for isotropic groups (=stabilizers) and isotropic quadratic forms.
The geometric intuition for the name isotropic group is motivated plausibly Here by Matt. About the usage of word 'isotropic' for isotropic quadratic forms I'm not sure.
Is there any geometric motivation for the terminology 'anisotropic' used for algebraic linear group defined above? Somehow it's geometry should reflect the property that 'something' there NOT behaves "the same in all directions."
Does it have any geometrical motivation or is it just a 'name'? My ingenuous attempt to explain the name is to interpret the split torus $T_k \cong \mathbb{G}_m^n$ as the sphere analoga for algebraic groups and the sphere is something where 'all directions are geometrically coequal', i.e. there is no distinguished direction, so maybe the name isotropic.
And an anisotropic algebraic group by definition above not contains such guy. Is this exactly the original heuristic motivation for the name anisotropic in context of algebraic groups by it's founder? Jaques Tits?