Let \(E/k\) be the minimal splitting field of \(T\) and set \(R_0 = R_{E/k}(T_E)\), which is induced: \(\Theta_E\) acts trivially on \(X^*(T)\), so \(X^*(R_0) = \operatorname{Ind}_{\Theta_E}^{\Theta} X^*(T)\) is a permutation module (Weil Restriction of Tori).

(1) Since \(T\) is commutative, the norm morphism \(N : R_0 \to T\), given on points of \(k\)-algebras after base change to \(k_s\) by multiplying the coordinates of the product decomposition, is a homomorphism defined over \(k\); on \(k\)-points it is \(t \mapsto \prod_{\iota} \iota(t)\). On characters it is the map \[ N^* : X^*(T) \longrightarrow \operatorname{Ind}_{\Theta_E}^{\Theta} X^*(T), \qquad \chi \longmapsto \textstyle\sum_{c} c \otimes c^{-1}\chi \] summing over the cosets \(c \in \Theta/\Theta_E\). This is injective, so \(N\) is surjective. Its cokernel is torsion-free: if \(m\,x = N^*(\chi)\) for some \(x\) in the induced module and \(m \geq 1\), the coefficient of the trivial coset gives \(\chi = m\psi\) for some \(\psi \in X^*(T)\), whence \(m\,x = m\,N^*(\psi)\) and \(x = N^*(\psi)\). A saturated inclusion of the character lattice means the kernel \(\ker N\) has torsion-free character group \(\operatorname{coker}(N^*)\), i.e. is a torus (Subtori and Quotients via the Character Lattice). Iterating the construction on \(\ker N\) and composing gives the two-step exact sequence.

(2) Choose a finite \(\Theta\)-stable generating set of \(X^*(T)\) (a \(\Theta\)-orbit of a generating set is finite by continuity). The induced surjection of \(\Theta\)-modules \(P \twoheadrightarrow X^*(T)\) from the permutation module \(P\) on that set corresponds, by F-Tori Are Classified by Galois Modules, to a closed immersion \(T \hookrightarrow R^0\) into the induced torus \(R^0\) with \(X^*(R^0) = P\). The quotient \(R^0/T\) has character group \(\ker(P \to X^*(T))\), a subgroup of a lattice, hence torsion-free, so \(R^0/T\) is a torus.