ARITHMETICS OF JORDAN ALGEBRAS 53
(8) M'
-v(d )
( d j ^ n ? ))' + ?
-v(d )
(d2(jB0 n p ))' + ?
for some d. e £ , if rank L = rank L = 1. Note that M = $ n M' in all
1 U 1 *-
three c a s e s . If M c M an order of ^, M' c M' . Now M' c some maximal
-"-stable order. But since e., e M' 1 i n it is clear that the only
11

maximal ^-stabl e order containing M' is
L \
to c
...
©
?
P
...
P
®
P
.. .
P
/
in the first case ,
© . . . © ©
©i
in the second,
\V
© . . . © ©
\ ? •• * P 0 /
in the third. Since # nthi s maximal *-stabl e order = M, M must equal M
which is therefore maximal. This completes the proof of the sufficiency part
of Theorem 3. The necessit y is now immediate. It suffices to observe that
by the first part of Theorem 2 (which ha s already been shown) all maximal
orders of ^ can be obtained from lattice s satisfying the hypotheses of
Theorem 3, so M' must be of the form (5), (6), (7) or (8) (up to isomorphism).
Assume that the hypotheses are broken by a lattice L = L 1 L other than
the exception which has already shown not to yield maximal orders. At leas t
one of Lemmas 14, 16 or 17 with r 2 may be applied to a line of L and a
line of L
?
, a line of L. and a plane of L., i ^ j {1, 2} , or a plane of L
^
I
j i
and a plane of "L . So M' will not be of the form (5), (6), (7) or (8) and M
is therefore not maximal.
q. e. d.
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