Let us assume that $G$ is an anisotropic semisimple, connected algebraic group over a field $k$ of characteristic zero. Let $K_G$ denote the class of its Killing form in the Witt ring of $k$.

Let $X$ be the projective quadric defined by $K_G$. It is known to be smooth (i.e $K_G$ is non degenerate).

For every quadratic form, there is a tower of field extensions, starting with $k:=k_0$. Then $k_1:= k(X)$. Over $k(X)$, we have that $K_G$ becomes isotropic. We denote the quadric defined by its anisotropic kernel by $X_i$. We set $k_i:=k(X_i)$. Eventually $K_G$ will become hyberpolic, i.e. a sum of hyberbolic planes. If we write down the number of hyperbolic planes, split off in each step, we obtain the splitting pattern of $K_G$ (it is also called the relative Witt indexes).

Question: How does the Tits index of $G$ change, depending on the splitting pattern of $K_G$?

For groups of type $D_n$ (or lets say for SO$(q)$ and Spin$(q)$, with $q$ denoting some quadratic form), this question is not too interesting as the Killing form of $G$ is basically the quadratic trace form.

Question: How do things look for the exceptional cases?

$F_4, E_6, E_7$ or $E_8$ would be of my interest.

The background of this question is that this gives us some kind of intrinsic sequence of Tits indexes of $G$.

The Killing form usually has a high rank (i.e. its number of variables). Not much is known about the splitting patterns of quadratic forms of high rank.