ARITHMETICS OF JORDAN ALGEBRAS 45
(1') v(d.) = v(d.) for all d.,d . elementary .
The proof of Theorem 3 proceeds as follows. First the exception noted in the
statement is shown to be properly contained in another order of #. Then in a
serie s of lemmas M' (the o-order of & generated by M) is computed for
four low dimensional c a s e s , namely, two lines , a plane, a line and a plane ,
and two planes . Finally the general c a s e is obtained from t h e s e .
Suppose L is a s in the exception. Arguing a s in Theorem 1 we may
assum e that L is a s in the hypothesis with v(c ) = 1, v(c ) = 0,
1 i r and vfd.) = 0 for d. elementary. Let A = ( a
1
. ) = $ , a ,
1
= a . ,
J J U n' ij
ji'1
a.. & . Then HA
e
E(L) if and only if
d a . + c a , . s = 2 t - l , l t r ,
s sj s s+1 j '
(4) c a . + d ^ . a ,_ , O 1 j n .
s sj s+1 s+1 j
d,a., eO 1 j n, 2r i n .
I
ij - - -
Since v(d ) v(c ) and v(d ) v(c ) = v(c ), s = 2t - 1, 1 t r, (4) is
S S S
IJ.
S S
equivalent to
c a , _ .

© s = 2 t - l , l t r ,
s s+1 j ' *
(4') c a .

O 1 j n .
s sj
d.a.,

^n 1 j n, 2r i n .
i U - - -
Let D = diag{p, 1, . . . , 1 } . We claim that M C ^nE(LD) . The matrix of h
-1 ~ l l
re LD is D HD = H' . So H' is the same as H except for the first
plane which becomes
Previous Page Next Page