I'm currently self-studying Lie Algebras and I came across the definition of the Killing Form. As I understand it, the Killing Form gives you an inner product with which you can visualize the roots of a Lie Algebra. Two questions here:
The definition of the Killing Form seems very random. Is there a natural reason why someone would choose this particular inner product with which to visualize the fundamental roots? Is there really no simpler inner product to choose?
What deeper insight does the root system give you about the Lie Algebra? As an example, I've attached a screenshot of a sample root system below. My issue is that it's so many layers thick with abstraction (each point is an "eigenvalue of the action of Cartan Subalgebra under the adjoint map" -- gosh, even saying that makes my head spin!) that I can't get a grip of what the diagram is saying morally.
To sum up, where I'm at right now is this: "the eigenvalues of the adjoint map form a nice picture if we arrange them according to this seemingly random inner product (the Killing Form)." But why are the eigenvalues of the adjoint map significant, and why is their arrangement in the below diagram significant? I feel like I am missing the big picture. Any suggestions would be appreciated!
Let me try to explain your first point, about the origin and importance of the Killing form. If I have a break from my work I can try to go into the second point, or someone more of an expert in Lie algebras than I can do it first.
If $\mathfrak g$ is a simple Lie algebra, then there is a unique non-degenerate bilinear form on the adjoint representation of $\mathfrak g$. This is a general fact about simple modules, and simply comes from the fact that the adjoint representation is self-dual, so there is a unique map $V\otimes V\to k$. (I chose $V$ and $k$ here because this is a general statement about self-dual simple modules over some object and a field $k$, be they Lie algebras, algebraic groups, etc.)
It turns out that the map is symmetric (i.e., comes from a map from the symmetric square of the adjoint, rather than the exterior square). So the reason for the definition in one sense is that the Killing form is unique, and that is it.
If one takes a step back, and looks at the theory of finite-dimensional $k$-algebras, then one encounters (nowadays, certainly not in 1910) the idea of a symmetric algebra. This is a $k$-algebra with a symmetric bilinear form satsifying $(ab,c)=(a,bc)$. The Killing form satisfies this relation also. So the Killing form is trying to turn the Lie algebra into a symmetric algebra. Now normally symmetric algebras are associative, but we'll not worry about this.
What do symmetric bilinear forms look like? They are often called symmetrizing trace forms, and we start to see the first connections with the definition of a Killing form. It turns out that this is the usual way to define symmetrizing trace forms, they come from trace maps. Indeed, the symmetrizing form on a matrix algebra is simply the trace map.
So not only is the Killing form the only way to define it, it is the standard way to define such a map.