Isomorphism theorem for complex semisimple Lie algebras

Given: Finite-dimensional complex semisimple Lie algebras $\mathfrak{g},{\mathfrak{g}}^{\mathrm{\prime }}$ with Cartan subalgebras $\mathfrak{h},{\mathfrak{h}}^{\mathrm{\prime }}$, root systems $\mathrm{\Phi },{\mathrm{\Phi }}^{\mathrm{\prime }}$ and bases $\mathrm{\Delta },{\mathrm{\Delta }}^{\mathrm{\prime }}$ whose Cartan matrices are equal, $A=A^{\mathrm{\prime }}$.

AC is assumed; it is used through the Serre presentation theorem and the root-system theorem (The Axiom of Choice).

The root system of a complex semisimple Lie algebra is a reduced crystallographic root system, and the Cartan matrix of a base is $a_{ij}={\alpha }_j(h_i)$ (Roots of a complex semisimple Lie algebra form a reduced crystallographic root system, Cartan matrix of a based root system).

Every root $\alpha$ admits elements $e_{\alpha }\in {\mathfrak{g}}_{\alpha }$ and $f_{\alpha }\in {\mathfrak{g}}_{-\alpha }$ such that $(e_{\alpha },f_{\alpha },h_{\alpha })$ is a root ${\mathfrak{s}\mathfrak{l}}_2$ triple (The root ${\mathfrak{s}\mathfrak{l}}_2$ triple).

Once root ${\mathfrak{s}\mathfrak{l}}_2$ triples have been chosen for the simple roots, every finite-dimensional complex semisimple Lie algebra with Cartan matrix $A$ is isomorphic to the Serre algebra $\mathfrak{g}(A)$ via its canonical generators (Serre presentation theorem).

Two based root systems with equal Cartan matrices are isomorphic by the map carrying corresponding simple roots to one another (The Cartan matrix determines a based root system).