Kac-Moody algebras/groups for reductive groups 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
$$ A = \begin{pmatrix} 2 &amp; -1 \\ -1 &amp; 2 \end{pmatrix} $$
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
$$\begin{aligned}
\text{coweight lattice} = Y &amp;= \mathbb{Z}\{\varepsilon_1^\vee, \ldots, \varepsilon_n^\vee\}, \\
\text{weight lattice} = X &amp;= \mathbb{Z}\{\varepsilon_1, \ldots, \varepsilon_n\}, \\
\text{perfect pairing}: \quad&amp; \langle \varepsilon_i^\vee, \varepsilon_j \rangle \mapsto \delta_{ij}, \\
\text{coroots} = \Pi^\vee &amp;= \{\alpha_1^\vee = \varepsilon_1^\vee - \varepsilon_2^\vee, \ldots, \alpha_{n-1}^\vee = \varepsilon_{n-1}^\vee - \varepsilon_n^\vee \}, \\
\text{roots} = \Pi &amp;= \{\alpha_1 = \varepsilon_1 - \varepsilon_2, \ldots, \alpha_{n-1} = \varepsilon_{n-1} - \varepsilon_n \}.
\end{aligned}$$
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".
Joppy said in comment:
The answer is yes, you can use pretty much whatever realisation of the root data you want and have the Kac-Moody construction go through with minor changes (dimension of the centre, etc). The most important thing for the theory is that the simple roots and simple coroots each remain linearly independent: without this, significant parts of the theory change.
Joppy said in comment:
The answer is yes, you can use pretty much whatever realisation of the root data you want and have the Kac-Moody construction go through with minor changes (dimension of the centre, etc). The most important thing for the theory is that the simple roots and simple coroots each remain linearly independent: without this, significant parts of the theory change.