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.