A diagonalizable group defined over a field k is said to split over k or k-split if the isomorphism is defined over k. This coincides with the usual notion of split for an algebraic group.
The category of diagonalizable groups defined over k is equivalent to the category of finitely generated abelian groups with Gal(ks/k)-equivariant morphisms without p-torsion, if k is of characteristic p. This is an analog of Poincaré duality and motivated the terminology.
A diagonalizable k-group is said to be anisotropic if it has no nontrivial k-valued character.
The so-called "rigidity" states that the identity component of the centralizer of a diagonalizable group coincides with the identity component of the normalizer of the group.
The fact plays a crucial role in the structure theory of solvable groups.