Cartan subalgebras and Jordan Normal form I'm stuck with Kac notes on Introduction to Lie Algebras. I logically understand all the definitions and everything is fine but I can't understand what's the thinking behind it. So I'm not asking for a rigorous proof o explanation, while I'd like a picture of what's happening behind the scenes.
We have for semi-simple Lie Algebra that the algebra is decomposable in: 
$$\mathfrak{g}=\mathfrak{h}\oplus(\bigoplus_{\alpha \in \mathfrak{h}^*} \mathfrak{g}_\alpha)$$
Where $\mathfrak{h}$ is the Cartan subAlgebra, the $\alpha$ are the roots, etc...
I'd like to know:


*

*what's going on. Please pick the most useful point of view let you understand the picture (doesn't matter if it's geometric, algebric, heuristic, etc...)

*what is the relation if any with the Jordan Normal Form? (In plain english if possible)

 A: The "weight spaces" $\mathfrak{g}_\alpha$ are like the "generalized eigenspaces", which correspond to the blocks of a Jordan matrix. More specifically, for an element $x \in \mathfrak{h}$, it's acting on the whole space $\mathfrak{g}$ via the adjoint representation ($\mathrm{ad}_x(y) = [x,y]$). So you can think of $x$ (or $\mathrm{ad}_x$) as a matrix, after picking a basis for $\mathfrak{g}$. If $\alpha$ is an eigenvalue of $x$, then when you write $x$ in Jordan canonical form, it is block diagonal, and one of the blocks will look like:
$$ \left( \begin{array}{cccc} \alpha & 1 & & \\ 
 & \alpha & \ddots & \\
 & & \ddots & 1 \\
 & & & \alpha \end{array} \right) $$
The invariant subspace corresponding to this block of the matrix is the $\mathfrak{g}_\alpha$, the "generalized eigenspace". Notice the connection to the property which defines the $\mathfrak{g}_\alpha$: if $v \in \mathfrak{g}_\alpha$, there is some $k$ so that $(x - \alpha \, \mathrm{Id}_\mathfrak{g})^k v = 0$. Here $k$ is the size of the block corresponding to $\alpha$, and the matrix above certainly satisfies $(x - \alpha \, \mathrm{Id})^k = 0$.
