Root system independent of chosen Cartan algebra

Question

I have read on "Lectures on Lie groups and Lie algebras" (by Carter, Segal, Macdonald) that Cartan subalgebras are related by some automorphism of the Lie algebra and this is proved using a density argument and ideas from algebraic geometry. Can someone give me the idea behind this or the main arguments? Thanks

Answer 1

The result, which you mention is the theorem that all Cartan subalgebras over an algebraically closed field of characteristic zero are conjugated under the group of special Lie algebra automorphisms. The proof uses only very few arguments from "algebraic geometry". Over algebraically closed fields, Cartan subalgebras are given by regular elements $h\in L$. Here is an example, where such an argument is used:

Lemma: The subset $L^{reg}\subset L$ of regular elements in the Lie algebra $L$ is a non-empty Zariski-dense subset of $L$, which is invariant under all automorphisms of $L$.

One does not need much more than the definition of the Zariski topology, and vanishing of polynomial functions. Most books on Lie algebras give a proof of the conjugacy of Cartan subalgebras, where you can see the details.