The question in brief is: can $\mathfrak{gl}_n$ be constructed as a Kac-Moody algebra, and can $\operatorname{GL}_n$ be constructed as a Kac-Moody group in a compatible way? In general, given any root datum, can we construct the corresponding Lie algebra and reductive group?
Background: In the books I've read on Kac-Moody algebras and groups (Infinite Dimensional Lie Algebras, by Kac, and Kac-Moody Groups, their Flag Varieties and Representation Theory by Kumar), it seems that the option of constructing a reductive (not just semisimple) Lie algebra or group is ruled out, since the only input for the construction is a generalised Cartan matrix rather than a root datum. For example, all of the reductive groups $\operatorname{SL_3}, \operatorname{GL}_3, \operatorname{PSL}_3$ have the Cartan matrix $$ A = \begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix} $$ however they are all different reductive groups. Can these all be constructed as Kac-Moody groups associated to that Cartan matrix? Looking at the corresponding Kac-Moody algebra, we have $\mathfrak{g}(A) \cong \mathfrak{sl}_3 \cong \mathfrak{psl}_3$, however it seems that $\mathfrak{gl}_3$ cannot be constructed since these books require that the dimension of the Cartan subalgebra is $n + \operatorname{corank}(A) = 2$, while $\mathfrak{gl}_3$ has a three-dimensional subalgebra.
It seems that as input to the Kac-Moody construction, one could instead put a root datum. For example, to construct $\operatorname{GL}_n$ or $\mathfrak{gl}_n$, we would use the root datum $$\begin{aligned} \text{coweight lattice} = Y &= \mathbb{Z}\{\varepsilon_1^\vee, \ldots, \varepsilon_n^\vee\}, \\ \text{weight lattice} = X &= \mathbb{Z}\{\varepsilon_1, \ldots, \varepsilon_n\}, \\ \text{perfect pairing}: \quad& \langle \varepsilon_i^\vee, \varepsilon_j \rangle \mapsto \delta_{ij}, \\ \text{coroots} = \Pi^\vee &= \{\alpha_1^\vee = \varepsilon_1^\vee - \varepsilon_2^\vee, \ldots, \alpha_{n-1}^\vee = \varepsilon_{n-1}^\vee - \varepsilon_n^\vee \}, \\ \text{roots} = \Pi &= \{\alpha_1 = \varepsilon_1 - \varepsilon_2, \ldots, \alpha_{n-1} = \varepsilon_{n-1} - \varepsilon_n \}. \end{aligned}$$ Then the matrix $a_{ij} = \langle \alpha_i^\vee, \alpha_j \rangle$ is the type $A_{n-1}$ Cartan matrix, and we should be able to define the Cartan subalgebra $\mathfrak{h} = Y \otimes_{\mathbb{Z}} \mathbb{C}$, making it $n$ (rather than $n-1$) dimensional, and identifying $\mathfrak{h}^*$ with $X \otimes_{\mathbb{Z}} \mathbb{C}$ by extending the perfect pairing up to $\mathbb{C}$. Would most of the theory of Kac-Moody algebras and groups go through if the construction allowed this? If not, what would fail? The most extreme example of this would be constructing the Lie algebra corresponding to a torus, which would simply be the $n$-dimensional abelian Lie algebra for any $n \geq 0$ fixed in advance.
Update: I should point out some of the reasons for asking this. Firstly, there is a very good off-the-self theory for Kac-Moody algebras, and it would be nice to know if this theory could be applied to any root datum (for example, to $\mathfrak{gl}_n$ and $\hat{\mathfrak{gl}_n}$). Secondly, it seems that some authors believe that Kac-Moody algebras are defined in this generality (or perhaps to them, the term "Kac-Moody algebra" means something more general). For example, in Lusztig's book Introduction to Quantum Groups, he refers to "the Kac-Moody Lie algebra attached to the root datum".
