52 M. L. RACINE

the exception is ruled out, we must have two such planes to which Lemma 17

(with r = 0) may be applied. The assumption rank L. ^ 1, j = 1,2 implies

that if L, has a line as summand it must have at leas t one other summand.

J

If this other summand is a plane apply Lemma 16 (with r = 0). Therefore

applying Lemmas 14-17 to L., j = 1, 2, (with r = 0) we get the two diagonal

blocks of (5). To obtain the two off-diagonal blocks apply Lemma 16 to a

line of L and a plane of L„ k ^ j e { 1 , 2 } , (with r = 1) or Lemma 17 to a

k j

plane of L and a plane of L (again with r = 1). Since M* is a maximal

^-stabl e order and M = $ n M ' , M is maximal by Proposition 2.

The same type of argument yields

"(d(fl n ? " v ( d ) ) ) ' + ? © . . . 0

© . . . ©

(6) M' for some d e & ,

if rank L = L rank L ,

©

(7) M'

©

©

? .. . ? (d(*0 n

jTv(d)))'

+?

for some d €

0 '

if rank L = 1 rank L and