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 , . € r» 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