ARITHMETICS OF JORDAN ALGEBRAS 47

a a \

PROOF. Let A=f l l 1 2 \ ; H A

€

M if and only if

a a /

12 22/

-v(d.) ^ -v(d )

a . ^ i S n ? L and a

1 9

* ( d , d ) © = ? , where (d d ) is the

-v(d )

fractional ©-ideal generated by d, and d„. So d.($ n ? ) e . . C M ,

1 2 i 0 n 1

-v(d )

and (d (& n ? ))' e C M ' In particular e

€

M . Hence

1 0 U X _ l l 1 -v( d )

e H a [ 1 2 ] = d i a S 1 2 a n d e 2 2 a f l 2 ] = d 2 a e 2 1 € M i W h 8 n a ^ ' T h u S

• v ( d

)

(d (* n $ ))' +f D

M C I I C M ' But

r - V ( d 2 r

? (d2(fiQ n ? ))" + * P V

• v ( d

)

(d1(j9Q n 15 ))' + ? 0

' is a ring. Therefore it

-v(d )

f (d2(&Q n ? ))' +tf'

must equal M' .

q. e.d .

As will be see n in future examples even when the K-algebra generated by

d(& n I5 ) = & the o-algebra generated by d($ n ? ) need not be P .

LEMMA 15. Let H = I , d. € & ; v(d ), v(d ) v(c),

\c dj

M = {H A| A

€

&(#

?

, $ ), H A e © }. Assume moreover that if & has no

symmetric prime then v(c) is even. Then M' = O .

'd c \/a 0\ / d a 0\

PROOF. We have [_ = ] . Therefore

c d

2

/ \ ° 0' \ c a 0