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
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