Let $\frak g$ be a semisimple Lie algebra over a finite dimensional field $F$, and let $\frak h$ be a Cartan subalgebra in $\frak g$. I need actually some explanation on why $$\mathfrak g = \bigoplus_{\alpha\in \frak h^{*}} \mathfrak g_{\alpha} $$ Where $\mathfrak g_\alpha= \{x\in \mathfrak g; \ [x,h]=\alpha(h)x\text{ for all } h\in \mathfrak h\}$.


The proof relies on $F$ being algebraically closed. There are examples of Lie algebras which don't decompose like this when $F$ is not algebraically closed.

This is the viewpoint of Humphrey's Intro to Lie Algebras and Rep Theory. The big picture is: The set of matrices $\{\text{ad}(h):\ h\in\mathfrak h\}$ is a bunch of diagonalizable (aka semisimple) matrices which all commute with each other. When diagonalizable matrices commute then they can be simultaneously diagonalized.

This is basically because if $A$ and $B$ are commuting matrices, then the action of $B$ preserves the $\lambda$-eigenspace for $A$. So you pick out a basis of eigenvectors for $B$ in the space of $\lambda$-eigenvectors for $A$ for every $\lambda$, then both $A$ and $B$ are diagonal with respect to this basis.

So if you are with me so far, then you are basically done — there is a basis of $\mathfrak g$ so that for any $h\in \mathfrak h$, the matrix of $\text{ad}(h)$ is diagonal. That is exactly what that decomposition means — each root space is simultaneously an eigenspace for each $h$ in $\mathfrak h$.

The fact that $\text{ad}(h)$ for $h\in\mathfrak h$ always semisimple is little bit subtle. The crux of it is that you can define what it means for an element $s$ in a semisimple Lie algebra to be `ad-semisimple', meaning that $\text{ad}(s)$ is a semisimple endomorphism (i.e. it is diagonalizable). In fact, every element in a semisimple Lie algebra can be decomposed into the sum of an ad-semisimple element with a ad-nilpotent element such that the two commute — and this is unique. This is called the abstract Jordan decomposition.

A toral subalgebra is defined to be an algebra containing only semisimple elements. A Cartan subalgebra is defined to be a maximal toral subalgebra. These are proven to exist and be nonzero in semisimple Lie algebras and all the properties of Cartan subalgebras can be derived from there.

It should be noted that this is just one approach to Cartan subalgebras. There are some equivalent definitions and different approaches to the theory.


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