706 7. ELIMINATING CASES CORRESPONDING TO NO SHADOWSuppose first that (HO) holds. Then W contains the orthogonal sum of the
hyperbolic 2-space Vo with w, and either w lies in Vi or Vi and hence is totally
singular, or W is diagonal and definite. Set w := vo+w for 0 =/. w E W, where
we choose v 0 to be singular in Vo n Vi-i in case W _ ::::; Vi, or the non-singular vector
in Vo in case W_ is definite. Then by construction w is singular and diagonal, soby H.4.2, w is 2-central, contrary to the previous paragraph. This establishes the
claim when (HO) holds.So suppose instead that (Hl) holds. As W contains no 2-central involution, W
is not Cv ( 02 (Li)), so 0 does not contain 2t. Therefore W is not centralized by
an involution of M-so that WE r in the language of Definition E.6.4. By (Hl),
· r > 2, so as m(W) = 3, W is maximal subject to Cc(W) i. M. But then E.6.11.2
says there is a subgroup of order 7 normal in NM (W), which cannot happen-
since NM (W) is a 7' -group unless W = Vi, where NM (W) ~ L3 (2) has no normal
subgroup of order 7. This completes the proof of the claim that V_ is faithful on
K.
Next observe that v_·induces inner automorphisms on K: We check that the
groups listed in Theorem C (A.2.3) have no A 4 -group of outer automorphisms,
whereas V_ = [V_, B]. Thus the projection VK of V_ on K is faithful of rank 4.
Let Z+ := 1 under (HO) and Z+ := Zi under (Hl). In either case, set fI :=
H/Z+· Now 02 (L+)Q is of index 2 in T+, and centralizes A= V+V_. Thus Acentralizes a subgroup of T+ of index 2. Therefore VK is centralized by QK :=
02(L+)QnK of index at most 2 in TK := T+ nK, so ZK := CvK(TK) is noncyclic
and contained in Z(TK)· Therefore m2(K/Z(K)) ;:::: 4 and Z(TK) is noncyclic.
We check the groups on the list of Theorem C for groups with these properties:
m 2 (K/Z(K)) 2 4 eliminates the groups in cases (1) and (2) of Theorem C (other
than As which also appears in case (4)), while Z(TK) noncyclic eliminates those in
cases (4) and (5) as well as those in case (3) over the field F 2. Therefore K/Z(K)
is of Lie type over F2n for some n > l. Now if R is a root group of K in TK,then 1 =/.Rn QK, so VK S CTK(R----n-Q"K) ::::; C7'K(R), and hence VK ::::; Z(TK),
so A centralizes TK. In particular m2(Z(TK)) 2 2, so either n 2 4 or K/Z(K) is
Sp4(4). Thus by I.1.3, the multiplier of K/Z(K) is of odd order, so as [A, TK] = 1,
[A, TK] ::::; Kn Z+ ::::; 02(K) = l. Therefore TK ::::; CT(A) = Q, so Q is Sylow in
QKo. However C(G, Q) ::::; M, so C(Ko, Q) ::::; Kon M < K 0 • Thus we may apply
the local C( G, T)-theorem C.l.29 to the maximal parabolics of K 0. Now if K is of
Lie type G2,^3 D4, or^2 F4, neither of the two maximal parabolics of K are blocks,so by C.l.29, each is contained in M. Thus K ::::; M as K is generated by these
maximal parabolics, a contradiction. This reduces us to the case where K/Z(K) is
a Bender group over F2n, L3(2n), or Sp4(2n), and MnKo is either a Borel subgroup
of Ko or a maximal parabolic Ki of K ~ L 3 (2n) or Sp 4 (2n). In any case Mn Ko
contains a Borel subgroup B of Ko normalizing TK. By an earlier remark, either
n 2 4 or K ~ Sp4(4).
Now let Y be a Cartan subgroup of B. By 7.7.4, Ye := Cy(V) is of index
at most 3 in Y. But when n 2 4, certainly IY : Yel > 3, since Ye centralizes
V and hence centralizes VK S Z(TK), while some subgroup of Y isomorphic to
Z2n-1 is semiregular on Z(TK)· Therefore Ko is Sp 4 (4), with Ye of order 3-again
centralizing V and hence VK. This is impossible, as the Cartan group of B is