Complete reducibility of rational modules in characteristic zero

(b)$\Rightarrow$(c). By [F4] there is a faithful finite-dimensional rational representation of $G$; if (b) holds, that representation is semisimple, so (c) holds.

(c)$\Rightarrow$(a). Let $V$ be a faithful semisimple finite-dimensional representation and put $U=R_u(G)$, which is unipotent and normal in $G$. For every simple subrepresentation $S\subseteq V$ the fixed space $S^U$ is nonzero by [F5]; it is a $G$-subrepresentation because $U$ is normal, so simplicity gives $S^U=S$, that is, $U$ acts trivially on $S$. Hence $U$ acts trivially on $V$, and faithfulness forces $U=1$. Over the perfect field $k$ of characteristic $0$, triviality of the unipotent radical is exactly reductivity of $G$, so (a) holds.

(a)$\Rightarrow$(b), reduce to a split central torus. Extend scalars to an algebraic closure $\bar{k}$; it is enough first to prove that $V_{\bar{k}}$ is semisimple. Apply [F1] to $G_{\bar{k}}$: its largest central torus $Z(G_{\bar{k}}{)}_t$ is a split torus, and its derived subgroup $G_{\bar{k}}^{\mathrm{\prime }}$ is semisimple. By [F8], $V_{\bar{k}}$ is the direct sum of the character weight spaces for $Z(G_{\bar{k}}{)}_t$. Since this torus is central in $G_{\bar{k}}$, each weight space is stable under $G_{\bar{k}}^{\mathrm{\prime }}$.

(a)$\Rightarrow$(b), decompose the weight spaces. Each weight space from step 1.3 is a finite-dimensional rational representation of $G_{\bar{k}}^{\mathrm{\prime }}$, so [F3] decomposes it into simple $G_{\bar{k}}^{\mathrm{\prime }}$-submodules. The central torus acts on each whole weight space by its character, so every such simple submodule is stable under both $Z(G_{\bar{k}}{)}_t$ and $G_{\bar{k}}^{\mathrm{\prime }}$, hence under their product $G_{\bar{k}}$. Thus $V_{\bar{k}}$ is semisimple. By [F7] semisimplicity descends from $\bar{k}$ to $k$, proving (a)$\Rightarrow$(b) over the original field.

Steps 1.1, 1.2 and 2.1 prove the equivalence of (a), (b) and (c). If (b) holds, an arbitrary rational representation $V$ is the union of its finite-dimensional subrepresentations by [F9], each of which is a direct sum of simple subrepresentations; the sets of simple subrepresentations whose sum is direct form a nonempty poset under inclusion. The union of a chain is again such a family, since every finite relation occurs in one chain member. By [A1] choose a maximal family with sum $S\subseteq V$. If $S\mathrm{\ne }V$, [F9] gives a finite-dimensional subrepresentation $W$ containing a vector outside $S$. A simple summand $C$ of $W$ is then not contained in $S$, and simplicity gives $C\cap S=0$, so adjoining $C$ extends the family, a contradiction. Hence $V=S$ is a direct sum of simple representations, that is, $G$ is linearly reductive.