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
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