1012 14. Ls(2) IN THE FSU, AND L 2 (2) WHEN .Cf(G, T) IS EMPTYin the remaining maximal 2-local D := (TX2,Xg) ~ S5/Ern/Zg of H over T,
which does not centralize i/2. As Xg centralizes i/2, Xg ¢. Y.
Let UD :=\VP); then 02 (D*) centralizes UD, and UD is the 6-dimensional ir-
reducible for D+ := D* /0 2 (D*) ~ S 6. Now ii has the structure of a 6-dimensional
F 4 -space preserved by K*, with the F 4 -points the irreducibles for Xi. This F 4-space structure restricts to UD of F 4 -dimension 3, and Vx 1 is the F4-point con-
taining i/2, so that Vx 1 ~ Es. Further T* X]'X2 is the stabilizer of an F4-line
Uo of UD containing Vx1' with X 1 X 2 T/02(X1X2T) ~ Sg x Sg. In particular X2
is fixed-point-free on Uo, so Vx 2 ~ Es. Thus to complete the proof, it remains
to show that Vx; s E for i = 1, 2. Now UD is totally singular, since UD is not
self-dual as a D-module. Thus UD s Vl = W, so by 12.8.11.1 it suffices to
show Vxi s [UD, Wg]. But there is x E wg inverting xt with x+ centralizing
Xi. As x+ inverts xt, Cun(x) = [UD,x] is of rank 3, and V2 s Cun(x). Then
Vx 2 = [i/2,X2] s [UD,x] s [UD, Wg], as required. As Wg* n K* induces a group
of F4-transvections on UD with center Vx 1 , Vx 1 s [UD, Wg* n K*] s E. This
completes the proof. DBy 14.4.9-14.4.12, we have reduced to the situation where one of cases (4)-(6)
·of of 14.4.2 holds, and in case (5) the chief factors for H^00 on ii are £ 2 (4)-modules.
By 14.4.8.1, L1 E Y; hence 14.4.10-14.4.12 show that in each case Y = {£ 1 ,X} is
of order 2, with XL1T/02(XL1T) ~ Sg x Sg.
Let Hl :=LT, H2 := L1XT, and Hg := LxT. Set F := {H1,H2,Hg} and
Go:= (F).
LEMMA 14.4.13. Go~ £4(2).PROOF. We show that (Go,F) is an Ag-system as defined in section I.5. Then
the lemma follows from 'L'heorem I.5.1. We just observed that H 2 /0 2 (H2) ~ Sg x Sg
and Hif02(Hi) ~ Lg(2) for i = 1, 3 by 14.4.8.2, so (Dl) and (D2) hold. As L 2 T is
maximal in Hl and Hg but X # £ 1 , L 2 T = H 1 n Hg, so £ 2 = L n Hg ::::1 Mn Hg
and hence L2T =Mn Hg. Thus as M = !M(LT), 02 (G 0 ) = 1, g so hypothesis
(D4) holds. Similarly L1T = Hl n H 2 and XT = H 2 n Hg, so (D3) holds. Finally(D5) is vacuous for a system of type Ag. D
We are now in a position to obtain a contradiction to our assumption that
d > 4. Namely as !Tl ~ IUI > 29 , Go is not £4(2), contrary to 14.4.13. This
contradiction shows:THEOREM 14.4.14. Assume Hypothesis 14.3.1 holds with (V^01 ) nonabelian.
Then L/02(L) ~ Lg(2) and G is isomorphic to HS or G 2 (3).PROOF. By assumption, Hypothesis 14.3.10 holds. Thus L/0 2 (L) ~ Lg(2) by
Theorem 14.3.16. Then by Theorem 14.3.26, either U = (Vi^01 ) is extraspecial orG ~HS, and we may assume the former. Hence if d = m(U) = 4, then G ~ G 2 (3)
by Theorem 14.4.3. Finally we just obtained a contradiction under the assumption
that U is extraspecial and d > 4, so the proof of Theorem 14.4.14 is complete. DgThe group J4 has the involution centralizer appearing in case (6) of 14.4.2, and there is
LE L1(G, T) with L/02(L) 3'! L3(2), but the condition 02(Go) = 1 fails as L rf= Lj(G, T).