The theorem states that: given a root system ${\displaystyle \Phi }$  in a Euclidean space with an inner product ${\displaystyle (,)}$ , ${\displaystyle \langle \beta ,\alpha \rangle =2(\alpha ,\beta )/(\alpha ,\alpha ),\beta ,\alpha \in E}$  and a base ${\displaystyle \{\alpha _{1},\dots ,\alpha _{n}\}}$  of ${\displaystyle \Phi }$ , the Lie algebra ${\displaystyle {\mathfrak {g}}}$  defined by (1) ${\displaystyle 3n}$  generators ${\displaystyle e_{i},f_{i},h_{i}}$  and (2) the relations

${\displaystyle [h_{i},h_{j}]=0,}$ 

${\displaystyle [e_{i},f_{i}]=h_{i},\,[e_{i},f_{j}]=0,i\neq j}$ ,

${\displaystyle [h_{i},e_{j}]=\langle \alpha _{i},\alpha _{j}\rangle e_{j},\,[h_{i},f_{j}]=-\langle \alpha _{i},\alpha _{j}\rangle f_{j}}$ ,

${\displaystyle \operatorname {ad} (e_{i})^{-\langle \alpha _{i},\alpha _{j}\rangle +1}(e_{j})=0,i\neq j}$ ,

${\displaystyle \operatorname {ad} (f_{i})^{-\langle \alpha _{i},\alpha _{j}\rangle +1}(f_{j})=0,i\neq j}$ .

is a finite-dimensional semisimple Lie algebra with the Cartan subalgebra generated by ${\displaystyle h_{i}}$ 's and with the root system ${\displaystyle \Phi }$ .

The square matrix ${\displaystyle [\langle \alpha _{i},\alpha _{j}\rangle ]_{1\leq i,j\leq n}}$  is called the Cartan matrix. Thus, with this notion, the theorem states that, give a Cartan matrix A, there exists a unique (up to an isomorphism) finite-dimensional semisimple Lie algebra ${\displaystyle {\mathfrak {g}}(A)}$  associated to ${\displaystyle A}$ . The construction of a semisimple Lie algebra from a Cartan matrix can be generalized by weakening the definition of a Cartan matrix. The (generally infinite-dimensional) Lie algebra associated to a generalized Cartan matrix is called a Kac–Moody algebra.

The proof here is taken from (Serre 1966, Ch. VI, Appendix.) harv error: no target: CITEREFSerre1966 (help) and (Kac 1990, Theorem 1.2.). Let ${\displaystyle a_{ij}=\langle \alpha _{i},\alpha _{j}\rangle }$  and then let ${\displaystyle {\widetilde {\mathfrak {g}}}}$  be the Lie algebra generated by (1) the generators ${\displaystyle e_{i},f_{i},h_{i}}$  and (2) the relations:

• ${\displaystyle [h_{i},h_{j}]=0}$ ,
• ${\displaystyle [e_{i},f_{i}]=h_{i}}$ , ${\displaystyle [e_{i},f_{j}]=0,i\neq j}$ ,
• ${\displaystyle [h_{i},e_{j}]=a_{ij}e_{j},[h_{i},f_{j}]=-a_{ij}f_{j}}$ .

Let ${\displaystyle {\mathfrak {h}}}$  be the free vector space spanned by ${\displaystyle h_{i}}$ , V the free vector space with a basis ${\displaystyle v_{1},\dots ,v_{n}}$  and ${\textstyle T=\bigoplus _{l=0}^{\infty }V^{\otimes l}}$  the tensor algebra over it. Consider the following representation of a Lie algebra:

${\displaystyle \pi :{\widetilde {\mathfrak {g}}}\to {\mathfrak {gl}}(T)}$ 

given by: for ${\displaystyle a\in T,h\in {\mathfrak {h}},\lambda \in {\mathfrak {h}}^{*}}$ ,

• ${\displaystyle \pi (f_{i})a=v_{i}\otimes a,}$ 
• ${\displaystyle \pi (h)1=\langle \lambda ,\,h\rangle 1,\pi (h)(v_{j}\otimes a)=-\langle \alpha _{j},h\rangle v_{j}\otimes a+v_{j}\otimes \pi (h)a}$ , inductively,
• ${\displaystyle \pi (e_{i})1=0,\,\pi (e_{i})(v_{j}\otimes a)=\delta _{ij}\alpha _{i}(a)+v_{j}\otimes \pi (e_{i})a}$ , inductively.

It is not trivial that this is indeed a well-defined representation and that has to be checked by hand. From this representation, one deduces the following properties: let ${\displaystyle {\widetilde {\mathfrak {n}}}_{+}}$  (resp. ${\displaystyle {\widetilde {\mathfrak {n}}}_{-}}$ ) the subalgebras of ${\displaystyle {\widetilde {\mathfrak {g}}}}$  generated by the ${\displaystyle e_{i}}$ 's (resp. the ${\displaystyle f_{i}}$ 's).

• ${\displaystyle {\widetilde {\mathfrak {n}}}_{+}}$  (resp. ${\displaystyle {\widetilde {\mathfrak {n}}}_{-}}$ ) is a free Lie algebra generated by the ${\displaystyle e_{i}}$ 's (resp. the ${\displaystyle f_{i}}$ 's).
• As a vector space, ${\displaystyle {\widetilde {\mathfrak {g}}}={\widetilde {\mathfrak {n}}}_{+}\bigoplus {\mathfrak {h}}\bigoplus {\widetilde {\mathfrak {n}}}_{-}}$ .
• ${\displaystyle {\widetilde {\mathfrak {n}}}_{+}=\bigoplus _{0\neq \alpha \in Q_{+}}{\widetilde {\mathfrak {g}}}_{\alpha }}$  where ${\displaystyle {\widetilde {\mathfrak {g}}}_{\alpha }=\{x\in {\widetilde {\mathfrak {g}}}|[h,x]=\alpha (h)x,h\in {\mathfrak {h}}\}}$  and, similarly, ${\displaystyle {\widetilde {\mathfrak {n}}}_{-}=\bigoplus _{0\neq \alpha \in Q_{+}}{\widetilde {\mathfrak {g}}}_{-\alpha }}$ .
• (root space decomposition) ${\displaystyle {\widetilde {\mathfrak {g}}}=\left(\bigoplus _{0\neq \alpha \in Q_{+}}{\widetilde {\mathfrak {g}}}_{-\alpha }\right)\bigoplus {\mathfrak {h}}\bigoplus \left(\bigoplus _{0\neq \alpha \in Q_{+}}{\widetilde {\mathfrak {g}}}_{\alpha }\right)}$ .

For each ideal ${\displaystyle {\mathfrak {i}}}$  of ${\displaystyle {\widetilde {\mathfrak {g}}}}$ , one can easily show that ${\displaystyle {\mathfrak {i}}}$  is homogeneous with respect to the grading given by the root space decomposition; i.e., ${\displaystyle {\mathfrak {i}}=\bigoplus _{\alpha }({\widetilde {\mathfrak {g}}}_{\alpha }\cap {\mathfrak {i}})}$ . It follows that the sum of ideals intersecting ${\displaystyle {\mathfrak {h}}}$  trivially, it itself intersects ${\displaystyle {\mathfrak {h}}}$  trivially. Let ${\displaystyle {\mathfrak {r}}}$  be the sum of all ideals intersecting ${\displaystyle {\mathfrak {h}}}$  trivially. Then there is a vector space decomposition: ${\displaystyle {\mathfrak {r}}=({\mathfrak {r}}\cap {\widetilde {\mathfrak {n}}}_{-})\oplus ({\mathfrak {r}}\cap {\widetilde {\mathfrak {n}}}_{+})}$ . In fact, it is a ${\displaystyle {\widetilde {\mathfrak {g}}}}$ -module decomposition. Let

${\displaystyle {\mathfrak {g}}={\widetilde {\mathfrak {g}}}/{\mathfrak {r}}}$ .

Then it contains a copy of ${\displaystyle {\mathfrak {h}}}$ , which is identified with ${\displaystyle {\mathfrak {h}}}$  and

${\displaystyle {\mathfrak {g}}={\mathfrak {n}}_{+}\bigoplus {\mathfrak {h}}\bigoplus {\mathfrak {n}}_{-}}$ 

where ${\displaystyle {\mathfrak {n}}_{+}}$  (resp. ${\displaystyle {\mathfrak {n}}_{-}}$ ) are the subalgebras generated by the images of ${\displaystyle e_{i}}$ 's (resp. the images of ${\displaystyle f_{i}}$ 's).

One then shows: (1) the derived algebra ${\displaystyle [{\mathfrak {g}},{\mathfrak {g}}]}$  here is the same as ${\displaystyle {\mathfrak {g}}}$  in the lead, (2) it is finite-dimensional and semisimple and (3) ${\displaystyle [{\mathfrak {g}},{\mathfrak {g}}]={\mathfrak {g}}}$ .

• Kac, Victor (1990). Infinite dimensional Lie algebras (3rd ed.). Cambridge University Press. ISBN 0-521-46693-8.
• Humphreys, James E. (1972). Introduction to Lie Algebras and Representation Theory. Berlin, New York: Springer-Verlag. ISBN 978-0-387-90053-7.
• Serre, Jean-Pierre (1966). Algèbres de Lie semi-simples complexes [Complex Semisimple Lie Algebras]. Translated by Jones, G. A. Benjamin. ISBN 978-3-540-67827-4.