[proofplan]
We apply the finite-type Serre presentation theorem to the generalized Cartan matrix $A$. That theorem identifies the Lie algebra defined by the Chevalley-Serre generators and relations with the semisimple complex Lie algebra attached to the finite root system whose Cartan matrix is $A$. The identification gives finite-dimensionality, semisimplicity, the Cartan subalgebra generated by the images of the $h_i$, and the stated root system at the same time.
[/proofplan]
[step:Apply the finite-type Serre presentation theorem to the Cartan matrix $A$]
Let $A=(a_{ij})_{1 \leq i,j \leq n}$ be the finite type Cartan matrix from the theorem statement, and let $\mathfrak g(A)$ be the complex Lie algebra given by the Serre presentation with generators $e_i,f_i,h_i$ for $1 \leq i \leq n$ and the Chevalley-Serre relations determined by $A$. Since $A$ is of finite type, the finite-type Serre presentation theorem applies: the Lie algebra defined by these generators and relations is isomorphic to the finite-dimensional semisimple complex Lie algebra attached to the finite root system with Cartan matrix $A$.
[guided]
The hypothesis that $A$ is of finite type is the hypothesis that allows the Serre presentation to produce a finite-dimensional semisimple Lie algebra rather than an infinite-dimensional Kac-Moody algebra. We apply the finite-type Serre presentation theorem to the Cartan matrix $A=(a_{ij})_{1 \leq i,j \leq n}$. Its input is exactly a finite type Cartan matrix together with the Lie algebra defined from the Chevalley-Serre generators $e_i,f_i,h_i$ and relations determined by the entries $a_{ij}$. Its conclusion is that this presented Lie algebra is isomorphic to the semisimple complex Lie algebra associated to the finite root system whose Cartan matrix is $A$.
[/guided]
[/step]
[step:Read finite-dimensionality and semisimplicity from the Serre identification]
By the isomorphism supplied in the previous step, $\mathfrak g(A)$ is isomorphic to a finite-dimensional semisimple complex Lie algebra. Finite-dimensionality and semisimplicity are invariant under Lie algebra isomorphism, so $\mathfrak g(A)$ is finite-dimensional and semisimple.
[/step]
[step:Identify the Cartan subalgebra generated by the images of the $h_i$]
Let $\bar h_i \in \mathfrak g(A)$ denote the image of the Serre generator $h_i$ for $1 \leq i \leq n$, and define
\begin{align*}
\mathfrak h := \operatorname{span}_{\mathbb C}\{\bar h_1,\dots,\bar h_n\} \subset \mathfrak g(A).
\end{align*}
Under the finite-type Serre identification, this subspace corresponds to the standard Cartan subalgebra of the associated semisimple Lie algebra. Therefore $\mathfrak h$ is a Cartan subalgebra of $\mathfrak g(A)$.
[guided]
We must specify which elements span the asserted Cartan subalgebra. For each $1 \leq i \leq n$, let $\bar h_i$ be the image in the quotient Lie algebra $\mathfrak g(A)$ of the Serre generator $h_i$. Define
\begin{align*}
\mathfrak h := \operatorname{span}_{\mathbb C}\{\bar h_1,\dots,\bar h_n\} \subset \mathfrak g(A).
\end{align*}
The finite-type Serre presentation theorem identifies the presented Lie algebra with the semisimple Lie algebra attached to the finite root system of Cartan matrix $A$, and under this identification the span of the $\bar h_i$ is carried to the standard Cartan subalgebra. Since being a Cartan subalgebra is preserved by Lie algebra isomorphism, $\mathfrak h$ is a Cartan subalgebra of $\mathfrak g(A)$.
[/guided]
[/step]
[step:Identify the root system and its Cartan matrix]
Let $\Phi(\mathfrak g(A),\mathfrak h)$ denote the root system of $\mathfrak g(A)$ relative to $\mathfrak h$. The finite-type Serre presentation theorem identifies this root system with the finite root system whose simple roots have Cartan matrix $A$. Hence the root system of $\mathfrak g(A)$ relative to $\mathfrak h$ is precisely the finite root system with Cartan matrix $A$.
[guided]
The remaining assertion concerns the root decomposition relative to $\mathfrak h$. Let $\Phi(\mathfrak g(A),\mathfrak h)$ denote the set of nonzero weights for the adjoint action of $\mathfrak h$ on $\mathfrak g(A)$; this is the root system of $\mathfrak g(A)$ relative to $\mathfrak h$. The same Serre identification used above identifies the simple root data obtained from the generators $e_i,f_i,h_i$ with the finite root system whose Cartan matrix is $A$. Consequently the Cartan integers of the simple roots are exactly the entries $a_{ij}$ of $A$, and no additional roots occur. Therefore $\Phi(\mathfrak g(A),\mathfrak h)$ is the finite root system with Cartan matrix $A$.
[/guided]
[/step]
