By F-Tori Are Classified by Galois Modules, tori correspond anti-equivalently to \(\Theta\)-lattices. A \(k\)-homomorphism \(f : T \to T'\) is an isogeny iff \(X^*(f) : X^*(T') \to X^*(T)\) is injective with finite cokernel, iff \(X^*(f) \otimes \mathbb{Q}\) is an isomorphism: surjectivity of \(f\) with finite kernel translates through the anti-equivalence to exactness of \(0 \to X^*(T') \to X^*(T) \to X^*(\ker f) \to 0\) with \(X^*(\ker f)\) finite. Hence isogenies become isomorphisms after \(\otimes\, \mathbb{Q}\), giving that the rational module is an isogeny invariant. Conversely, an isomorphism \(\phi : X^*(T')_{\mathbb{Q}} \to X^*(T)_{\mathbb{Q}}\) can be scaled by a positive integer so that \(N\phi(X^*(T')) \subseteq X^*(T)\) with finite index; \(N\phi\) is then a lattice map realizing an isogeny \(T \to T'\). This proves (1), and (2) follows since morphisms up to isogeny are \(\operatorname{Hom}\otimes\mathbb{Q}\) on lattices, i.e. morphisms of rational representations; semisimplicity is Maschke for the finite quotient through which \(\Theta\) acts.

Example. The split–anisotropic decomposition (Split-Anisotropic Decomposition of a Torus) is the isotypic decomposition isolating the trivial \(\mathbb{Q}[\Theta]\)-isotype: \(T\) is isogenous to \(T_d \times T_a\) with \(X^*(T_d)_{\mathbb{Q}}\) the trivial isotype and \(X^*(T_a)_{\mathbb{Q}}\) its complement. \(\square\)