ARITHMETICS OF JORDAN ALGEBRAS 49
v ( d
1
) , v ( d
2
) v(c), M
3
= {H
3
A|A €W(&3,&0), H A
e
^ } . Consider M^
defined by H as in the preceding lemma, as embedded in M . Let
r = v(d~) - v(c) if v(d ) v(c) cas e i), v(c) - v(d ) if v(c) _ v(d ) c a s e ii).
"pr f D \
Then M' = [ ? r "f O + M^ in cas e i),
-v(d_)
r r ~V[U3 r
^f f ( d
3
( &
0
n«P ) ) ' + ? /
pr pr pr
M' =1 V y f + in c a s e ii).
V -v(d ) '
n (d
3
(^
Q
n ? ))• + ? r y
-v(d )
PROOF. Arguing a s above (d ( 1 n ? ))'e c M' and in
-1 - s
particular e
3
M
3
^ a r
1 3
v a r
2 3
"i 6 M
3
if and only if a

(d , c) © = y ,
~s —s
where s = min(v(d ), v(c)). Multiplying ?r
13
-i and ?r
23
-i by e we obtain
\ 4. -s+v(d ) -s+v(d )
? " S + V ( C ) e
1 3
, ? " S ( C ) e
2 3
, ? 3 e
3 1
and ? 3 e
3 2
c M^. In c a s e i)
we have
£
O | + M' C M*
-v(d )
?r
f (d3(^Q n ? ))'/
In c a s e ii)
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