= Solution
For a compact <Lie group>, take any positive-definite <Hermitian inner product> $h_0$ on $V$ and average using probability <Haar measure>:
$$
h(v,w)=\int_G h_0(gv,gw)\,dg.
$$
It remains positive definite, because the integrand is positive for a nonzero vector, and translation invariance makes it $G$-invariant. If $W\subset V$ is an <invariant subspace>, then $W^\perp$ is invariant too: $h(gv,w)=h(v,g^{-1}w)=0$ when $v\in W^\perp$. Thus $V=W\oplus W^\perp$. Induction on dimension proves \b[complete reducibility]. This uses the usual continuous finite-dimensional representations of the <compact group>.
For a connected complex <reductive algebraic group>, representations are understood to be <rational representations>. Use a <compact real form> $K\subset G$ with $\operatorname{Lie}(G)=\operatorname{Lie}(K)\otimes_{\mathbb R}\mathbb C$. The structural construction takes a <compact real form> of the semisimple derived factor and the unit circles in the central torus, then passes through their finite central quotient. This is the compact-form structure of a complex reductive group, rather than an assumption that its given representation is already unitary.
The group $K$ is Zariski dense: the <Lie algebra> of its Zariski closure contains the complex span of $\operatorname{Lie}(K)$, hence all of $\operatorname{Lie}(G)$; connectedness then makes that closure $G$. Average over $K$ as above. The <orthogonal complement> of a $G$-<submodule> is $K$-invariant. Its stabilizer is a Zariski-closed subgroup, since preserving a fixed subspace is a system of algebraic matrix conditions. The stabilizer contains dense $K$, so it is all of $G$. This again gives an invariant complement to every <submodule> and proves
$$
\boxed{V\cong\bigoplus_j V_j\quad\text{with each }V_j\text{ irreducible}.}
$$
If reductivity is defined to allow finitely many components, first obtain a $G^0$-equivariant projection $P:V\to W$. Average its conjugates over the finite component group. Since $P$ commutes with $G^0$, the average is independent of coset representatives; it is $G$-equivariant and remains the identity on $W$. Its kernel supplies the required complement. Thus the result holds in that convention too.
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