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 + M» 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)