Cartan-Killing classification of complex simple Lie algebras

A simple $\mathfrak{g}$ is semisimple: its radical is an ideal, hence zero or all of $\mathfrak{g}$, while $[\mathfrak{g},\mathfrak{g}]=\mathfrak{g}$ by nonabelianness and simplicity, so its derived series cannot reach zero. Choose a Cartan and a base by [L1]. Its root set is nonempty: otherwise the root spanning property gives $\mathfrak{h}=0$ and then the root decomposition gives $\mathfrak{g}=0$, contrary to simplicity. If the Dynkin diagram were disconnected, split its indices into two nonempty blocks I,J. The subalgebras generated by the triples in each block commute: [L6] gives this for generators, and repeated Jacobi identities extend it to all bracket words. Their sum is a subalgebra containing all generators, hence all of $\mathfrak{g}$. Both subalgebras are nonzero ideals. They cannot both equal $\mathfrak{g}$, since then their commutation would make $\mathfrak{g}$ abelian; but simplicity would force just that. Thus the diagram is connected and $\mathrm{\Phi }$ irreducible.

An algebra isomorphism carries a Cartan subalgebra to a Cartan subalgebra, since nilpotence and the self-normalizer condition are invariant under isomorphism. It carries root spaces to the corresponding root spaces, preserving the Killing form since adjoint matrices are conjugated. By Cartan conjugacy in [L1] one may compare with any Cartan chosen in the target. Choices of positive systems give isomorphic based systems by [L5]. Thus the Dynkin type does not depend on these choices and is invariant under algebra isomorphism.

If two complex simple algebras have root systems of the same classified type, [L2] supplies an unbased root-system isomorphism f. The image f(Delta) of the first base is a base in the second system: f carries the defining regular vector and positive half-space to those of a positive system. By [L5], compose f with a target Weyl element carrying that positive system to the one whose base was chosen there. The resulting based-root-system isomorphism satisfies exactly the hypothesis of [L4], so the algebras are isomorphic.

For every diagram in the list, [L3] supplies a nonempty irreducible root system. Its realization in [L4] is a finite-dimensional complex simple algebra. In particular rank one gives A_1; B_2=C_2 is represented only by B_2 and D_3=A_3 only by A_3, as in [L2].

By step 1.1 and [L2], every simple algebra has a type in the displayed list. Step 1.2 makes type well-defined on isomorphism classes and ensures distinct types cannot be isomorphic; step 1.3 proves injectivity within a type; step 1.4 proves existence for every type. This establishes the asserted bijection. The zero and one-dimensional abelian algebras are excluded by the nonabelian simple convention of [L6].