Serre's theorem on semisimple Lie algebras

Remarks

The proof first studies the algebra with only relations 1 to 3, which splits as $\mathfrak{n}^- \oplus \mathfrak{h} \oplus \mathfrak{n}^+$ with $\mathfrak{n}^\pm$ free on the $e_i$ or the $f_i$. Relations 4 generate ideals there, and in the quotient the operators $\operatorname{ad} e_i$ and $\operatorname{ad} f_i$ are locally nilpotent. This makes the Weyl group act, which bounds the weights by $\Phi$ and gives finite dimension. Example: for $\Phi = A_1$ there is one generator of each kind, and the relations give $[h, e] = 2e$, $[h, f] = -2f$, $[e, f] = h$, so $\mathfrak{g} = \mathfrak{sl}_2(F)$. Relations 4 are needed: without them the algebra is infinite-dimensional as soon as $l \ge 2$, for example $(\operatorname{ad} e_1)^n (e_2) \ne 0$ for all $n$. Since every semisimple Lie algebra satisfies these relations for suitable generators, it is isomorphic to the algebra built from its own root system. Thus every root system arises from a semisimple Lie algebra, unique up to isomorphism, which gives the existence part of the root system of a semisimple Lie algebra correspondence.

Going deeper

For a student

The theorem gives a uniform construction of all simple Lie algebras, including $E_6, E_7, E_8, F_4, G_2$, without explicit matrices. Removing the finiteness from the Cartan matrix and keeping the same relations defines the Kac–Moody algebras, of which the affine Lie algebras are the next simplest case.