In part 4, we prove two theorems about pairs L, V in the Fundamental Setup
(3.2.1): In chapter 10, we show that L = L 0. Then in chapter 11, we show that L
is not of Lie type of Lie rank 2 over F 2 n for n > 1.
A counter example in chapter 10 is of the form Lo= LLt with t ET - NT(L)
and L/0 2 (L) isomorphic to L 2 (2n) or Sz(2n) with n > 1, or to L 3 (2). In the first
two cases, we can view L 0 /0 2 (L 0 ) as of Lie type of Lie rank 2 over F 2 n. Thus the
majority of the effort in part 4 is devoted to the elimination of cases in the FSU
where Lo is of Lie type and Lie rank 2 over F 2 n for some n > 1.
One of the main tools for treating such groups is the study of Cartan subgroups,
both of Lo and of HE 1i*(T,M): a Cartan subgroup of X :=Lo or His defined
to be a Hall 2'-subgroup of Nx(T n X).
The most difficult cases are those where the Cartan subgroup is small or trivial:
that is, when n = 2, or in chapter 10 when L ~ L 3 (2) is defined over F 2.