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 $k\mathrel{:=}k_0$. Then $k_1\mathrel{:=} k(X)$. Over $k(X)$, we have that $K_G$ becomes isotropic. We denote the quadric defined by its anisotropic kernel by $X_i$. We set $k_i\mathrel{:=}k(X_i)$. 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 index).
Question: How does the Tits index of $G$ change, depending on the Witt index of $K_G$?
For groups of type $D_n$ (or lets say for $\mathbf{SO}(q)$ and $\mathbf{Spin}(q)$, 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 interest to me.
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.
