I will follow Chapter 2 of this paper by Valloni. Let $V$ be a finitely generated free abelian group. An integral Hodge structure of weight $k$ on $V$ is a decomposition $$ V \otimes_\mathbb{Z} \mathbb{C} = \bigoplus_{p+q=k} V^{p,q}. $$ The Deligne torus is the real algebraic group $\mathbb{S} = R_{\mathbb{C}/\mathbb{R}}(\mathbb{G}_{m,\mathbb{C}})$. That is, $\mathbb{S}$ is the Weil restriction of the complex multiplicative group from $\mathbb{C}$ to $\mathbb{R}$. Given a representation $h:\mathbb{S}\to \operatorname{GL}(V)_\mathbb{R}$, such that $h_{\mathbb{R}}(t) = t^{-k}\operatorname{id}_V$ for all $t\in \mathbb{R}^*$, Valloni claims that we obtain a Hodge structure on $V$ by $$ V^{p,q} = \{v \in V_\mathbb{C} : h_\mathbb{C}(z_1,z_2)v = z_1^{-p}z_2^{-q}v \text{ for all } z_1,z_2\in\mathbb{C}^*\}. $$ I've found this claim in numerous places on the internet, but I haven't been able to find any proofs. In particular, I don't see why $V\otimes_\mathbb{Z}\mathbb{C}$ is the direct sum of these $V^{p,q}$. Apparently, all characters of $\mathbb{C}^*\times \mathbb{C}^*$ are of the form $\sigma_{p,q}(z_1,z_2) = z_1^{-p}z_2^{-q}$ for some $p,q$, so I suppose we are somehow decomposing the representation $V\otimes_\mathbb{Z}\mathbb{C}$ into a direct sum of irreps. I'm familiar with the representation theory of finite groups, in which case this sort of thing is justified by Maschke's Theorem, but I'm not sure how to deal with it in the case of the infinite group $\mathbb{C}^*\times \mathbb{C}^*$.
So, my question is: how do we prove that $V\otimes_\mathbb{Z}\mathbb{C}$ is actually the direct sum of the $V^{p,q}$ defined above?
$\Bbb C^\times \times \Bbb C^\times$ is a diagonalizable algebraic group, this is equivalent to the fact that every (algebraic) representation $W$ is a direct sum of the eigenspaces associated to the different characters. $W=\bigoplus_{\chi \in X(G)} V_\chi$.
For a proof, see Theorem 12.12 in Milne's Algebraic Groups.
More generally, in characteristic $0$ every finite-dimensional algebraic representation of a reductive group is a direct sum of irreducible subrepresentations. Most algebraic groups that one considers in practice are reductive.