Complete reducibility reduces to splitting codimension-one simple submodules

(b)$\Rightarrow$(c) is immediate: a simple subrepresentation of codimension one is in particular a subrepresentation of codimension one.

(a)$\Rightarrow$(b). Since $V$ is finite-dimensional, its direct-sum decomposition has finitely many simple summands. Induct on their number. The zero-summand case is immediate. Write $V=S\oplus V^{\mathrm{\prime }}$ with $S$ simple and $V^{\mathrm{\prime }}$ a sum of fewer simple subrepresentations, and let $q:V\to V^{\mathrm{\prime }}$ be the projection. For a subrepresentation $W\subseteq V$, simplicity gives either $W\cap S=S$ or $W\cap S=0$. In the first case, $W=S\oplus (W\cap V^{\mathrm{\prime }})$; induction splits $W\cap V^{\mathrm{\prime }}$ in $V^{\mathrm{\prime }}$, hence splits $W$ in $V$. In the second case, $q{\mathrm{\mid }}_W$ is injective and $q(W)$ is a subrepresentation of $V^{\mathrm{\prime }}$; induction gives $V^{\mathrm{\prime }}=q(W)\oplus C^{\mathrm{\prime }}$ for some subrepresentation $C^{\mathrm{\prime }}$. Set $C=S\oplus C^{\mathrm{\prime }}$. Every $v\in V$ differs from some $w\in W$ by an element of $C$, because the $q(W)$ component of $q(v)$ has a unique lift through the injective map $q{\mathrm{\mid }}_W$. If $w\in W\cap C$, then $q(w)\in q(W)\cap C^{\mathrm{\prime }}=0$, so $w\in W\cap S=0$. Thus $V=W\oplus C$ in this case as well.

(c)$\Rightarrow$(b), induction on $n=\dim V$. Assume (c) and let $W\subseteq V$ be a subrepresentation of codimension one in an $n$-dimensional $V$. If $W=0$ then $V=0\oplus V$; if $W$ is simple then (c) applies; so suppose $W\mathrm{\ne }0$ is not simple. Then $W$ has a nonzero proper subrepresentation, and a maximal proper subrepresentation $W^{\mathrm{\prime }}$ of $W$ exists and has $W\mathrm{/}{W}^{\mathrm{\prime }}$ simple, since any strictly increasing chain of proper subrepresentations of the finite-dimensional $W$ has length at most $\dim W$. The quotient $V\mathrm{/}{W}^{\mathrm{\prime }}$ has dimension $n-\dim W^{\mathrm{\prime }}<n$ when $W^{\mathrm{\prime }}\mathrm{\ne }0$, and $W\mathrm{/}{W}^{\mathrm{\prime }}$ is a simple subrepresentation of codimension one in $V\mathrm{/}{W}^{\mathrm{\prime }}$, so (c) gives $V\mathrm{/}{W}^{\mathrm{\prime }}=W\mathrm{/}{W}^{\mathrm{\prime }}\oplus V^{\mathrm{\prime }}\mathrm{/}{W}^{\mathrm{\prime }}$ for a subrepresentation $V^{\mathrm{\prime }}\subseteq V$ containing $W^{\mathrm{\prime }}$; then $\dim V^{\mathrm{\prime }}\mathrm{/}{W}^{\mathrm{\prime }}=1$, so $\dim V^{\mathrm{\prime }}=\dim W^{\mathrm{\prime }}+1$. If $W^{\mathrm{\prime }}$ is simple, (c) splits the pair $W^{\mathrm{\prime }}\subseteq V^{\mathrm{\prime }}$; if $W^{\mathrm{\prime }}$ is not simple, then $\dim V^{\mathrm{\prime }}<n$ and the induction hypothesis (b) applied to $V^{\mathrm{\prime }}$ splits the pair $W^{\mathrm{\prime }}\subseteq V^{\mathrm{\prime }}$. In both cases $V^{\mathrm{\prime }}=W^{\mathrm{\prime }}\oplus L$ for a one-dimensional subrepresentation $L$. Now $W\cap V^{\mathrm{\prime }}=W^{\mathrm{\prime }}$ because $W\mathrm{/}{W}^{\mathrm{\prime }}\cap V^{\mathrm{\prime }}\mathrm{/}{W}^{\mathrm{\prime }}=0$ in the direct sum $V\mathrm{/}{W}^{\mathrm{\prime }}=W\mathrm{/}{W}^{\mathrm{\prime }}\oplus V^{\mathrm{\prime }}\mathrm{/}{W}^{\mathrm{\prime }}$, so $W\cap L\subseteq W\cap V^{\mathrm{\prime }}=W^{\mathrm{\prime }}$ and $W\cap L\subseteq L$ give $W\cap L\subseteq W^{\mathrm{\prime }}\cap L=0$; and $\dim W+\dim L=(n-1)+1=n=\dim V$ with $W+L\subseteq V$, so $V=W\oplus L$.

(b)$\Rightarrow$(a), first the splitting property. Assume (b) and let $W\subseteq V$ be a subrepresentation of a finite-dimensional $V$. If $W=0$, it is already a direct summand, so assume $W\mathrm{\ne }0$. On the rational representation ${\mathrm{Hom}}_k(V,W)$ of [F2] consider the subrepresentations $V_1=\{f:f{\mathrm{\mid }}_W=a{\mathrm{id}}_W\text{ for some }a\in k\}$ and $W_1=\{f:f{\mathrm{\mid }}_W=0\}$; both are subrepresentations because $W$ is stable under $G$, and $V_1\mathrm{/}{W}_1\cong k$ has dimension one. By (b) applied to the pair $W_1\subseteq V_1$ there is a one-dimensional subrepresentation $L$ with $V_1=W_1\oplus L$. Every nonzero $f\in L$ satisfies $f{\mathrm{\mid }}_W=a{\mathrm{id}}_W$ with $a\mathrm{\ne }0$; by [F3] the one-dimensional representation $L$ is trivial, so $g\cdot f=f$ for all $R$-points $g$, that is, $f(v)=g\cdot f(g^{-1}v)$ for all $v\in V$, which after replacing $g$ by $g^{-1}$ says $f(g\cdot v)=g\cdot f(v)$: $f$ is a homomorphism of rational representations. Replacing $f$ by $a^{-1}f$ we may suppose $f{\mathrm{\mid }}_W={\mathrm{id}}_W$, and then $V=W\oplus \ker f$ because $f{\mathrm{\mid }}_W={\mathrm{id}}_W$ gives $W\cap \ker f=0$ and $v-f(v)\in \ker f$ for every $v$.

(b)$\Rightarrow$(a), conclusion. Under (b) every subrepresentation of every finite-dimensional $V$ is a direct summand by step 1.4. We prove by induction on $\dim V$ that such a $V$ is a direct sum of simple subrepresentations. For $V=0$ this is the empty sum. If $V\mathrm{\ne }0$, choose a nonzero subrepresentation $S\subseteq V$ of minimal dimension; it is simple, because a proper nonzero subrepresentation of $S$ would be a nonzero subrepresentation of $V$ of smaller dimension. By step 1.4, $V=S\oplus C$ with $\dim C<\dim V$, and every subrepresentation of $C$ is a subrepresentation of $V$, so the induction hypothesis applies to $C$ and exhibits it as a direct sum of simple subrepresentations; adjoining $S$ gives such a decomposition of $V$.

Steps 1.1, 1.2, 1.3, 1.4 and 2.1 prove (a)$\Rightarrow$(b)$\Rightarrow$(c) and (c)$\Rightarrow$(b)$\Rightarrow$(a), so the three conditions are equivalent.