Groups of multiplicative type are linearly reductive

By [F1] choose a finite Galois extension $K\mathrm{/}k$ with group $\mathrm{\Gamma }=\mathrm{Gal}(K\mathrm{/}k)$ such that $G_K$ is diagonalizable with $O(G_K)=K[M]$, $M=X^{\ast }(G_K)$ finitely generated, and the $\mathrm{\Gamma }$-action permutes the characters with $\sigma (e_{\chi })=e_{\sigma \chi }$. The base change $V_K=V{\otimes }_{k}K$ is a representation of $G_K$ equipped with a semilinear $\mathrm{\Gamma }$-action compatible with the comodule structure, and by [F2] the passage $V\mapsto V_K$ is an equivalence with the corresponding category of finite-dimensional semilinear $\mathrm{\Gamma }$-equivariant $G_K$-representations, with inverse $W\mapsto W^{\mathrm{\Gamma }}$.

By [F3] the representation $V_K$ decomposes as a direct sum $V_K={⨁}_{\chi \in M}(V_K{)}_{\chi }$ of eigenspaces for the distinct characters, and the $\mathrm{\Gamma }$-equivariance of the comodule structure gives $\sigma ((V_K{)}_{\chi })=(V_K{)}_{\sigma \chi }$ for $\sigma \in \mathrm{\Gamma }$, so the decomposition is permuted by $\mathrm{\Gamma }$. For each $\mathrm{\Gamma }$-orbit $O\subseteq M$ put $W_O={⨁}_{\chi \in O}(V_K{)}_{\chi }$; then $W_O$ is a $\mathrm{\Gamma }$-stable $G_K$-subrepresentation and $V_K={⨁}_O{W}_O$ is a sum over the finitely many orbits.

Fix an orbit $O$, choose $\chi \in O$, and put ${\mathrm{\Gamma }}_{\chi }=\{\sigma :\sigma \chi =\chi \}$ and $K_{\chi }=K^{{\mathrm{\Gamma }}_{\chi }}$. The weight space $(V_K{)}_{\chi }$ has a semilinear ${\mathrm{\Gamma }}_{\chi }$-action. Galois descent [F2] identifies it with $K{\otimes }_{K_{\chi }}E$ for the $K_{\chi }$-space $E=((V_K{)}_{\chi }{)}^{{\mathrm{\Gamma }}_{\chi }}$. Choose a basis of $E$. Each basis vector defines a ${\mathrm{\Gamma }}_{\chi }$-stable $K$-line in $(V_K{)}_{\chi }$. Translate that line by representatives of $\mathrm{\Gamma }\mathrm{/}{\mathrm{\Gamma }}_{\chi }$ and take their direct sum across the weights in $O$; the result is a $\mathrm{\Gamma }$-stable $G_K$-subrepresentation with one-dimensional weight spaces, and it is independent of the choice of representatives because the initial line is ${\mathrm{\Gamma }}_{\chi }$-stable. These orbit subrepresentations, one for each basis vector, decompose $W_O$. Each descends by [F2] to a simple $G$-module: a submodule after scalar extension is a sum of some of its distinct one-dimensional weight spaces, and $\mathrm{\Gamma }$-stability and transitivity on $O$ force either none or all. Thus the whole isotypic block need not be simple, but is a direct sum of copies of one simple module indexed by $O$.

Every finite-dimensional representation is therefore a direct sum of simple modules. For each orbit $O$ the construction with a one-dimensional $K_{\chi }$-space gives a simple module, and any simple module must have a single orbit and multiplicity one by step 3.1; different orbits have different scalar-extended weights. This gives the asserted classification. For an arbitrary rational $V$, project its coaction onto the direct sum of weight-coalgebra blocks for each Galois orbit; these finite-dimensional blocks descend from the spans of $e_{\chi }$, $\chi \in O$, and counit and coassociativity give a direct decomposition $V={⨁}_O{V}_O$. Finite Galois descent also holds for arbitrary semilinear spaces: for each vector, its finite orbit under the Galois group spans a finite-dimensional stable subspace, to which [F2] applies. This proves surjectivity of the canonical map from the scalar extension of invariants; a finite relation among invariant vectors lies in such a finite stable subspace, and finite-dimensional descent proves injectivity. Apply this argument to each weight space and its stabilizer subgroup. In each block choose a basis of the descended possibly infinite-dimensional space $E$ under AC; the construction of step3.1 then decomposes $V_O$ into copies of its orbit simple. Hence every rational representation is a direct sum of simples and $G$ is linearly reductive.