ARITHMETICS OF JORDAN ALGEBRAS 41

2

a.a.* ® 7\. and (a.a.* + a.a.*) ® A.A. = (a.a.* + (a.a,*)*) ® A.A.. Since both

1 1 i i J J i i J i J i J i J

terms

e

&_ ® P we have that the P subspac e of M(JB 8 P, *) spanned bynorms

o $

of & 0 P is $ ® P. Therefore it suffices to consider $ ® P. If * is of the

second kind & ® P s P 0 P and * is the exchange involution;

n n

H ( P

n

0 P ° , * ) = { ( a , a ) | a

€

P

n

) and since (a, a) = (a, 1 )(l, a) = (a, 1 )(a, 1 )*,

fl„ 8 P = H(& 8 P, *). If * is of the first kind, & ® P s P . If * induces

o n

the transpos e involution on P then a + a =a + a *, i ^ j , is a trac e and

hence belongs to $^ ® P; e.. = e..e.. = e,,e

( 1

* e &~ ® P and the subspac e

0 n ii u n n 0

& ® P - M($ ® P, *). If # induces the symplectic involution, n = 2m and we

may consider P- as (P0)m with (a.,)* = (a..), a., e P^, the standard

2m 2'

IJ

j i "

IJ

2'

involution of P

0

the split quaternions over P. As above a,. + a.. = a

( 1

+ a.,*

Z i j j i

IJ

ij

m m

J J J J

~

6*o

P. The rrth entry of (a..)(ail) = / a a = V n(a ), where n is

ij ji L rs rs L rs

s= l s= l

the quaternion norm of P . Therefore the diagonal entries of elements of

fl ® P are scalar . For a = (a U

€

P a = j _ ~ j j . Therefore if

char. $ £ 2 then $ ® P = H(JB S P, *) and if char. $ = 2,

M P 5 M(& ® P, *) . Assume char. $ = 2 and & c & a subspac e of

U(&, *) such that xfl x* C A , V x « « . Then $_ ® p is a subspac e of

1 1 1 $

} ( ( ^ P , *) such that X(JB ® P)x" c * 0 P , V x J ® P , Since

& 8 p c $ 8 p, a.. +a..

6

$ , ® P and there exist s a diagonal element

m

d = Y a., e J&, ® P with d / A ® P. Therefore a., / P for some a., e P

0

;

. ^ n 1 0

JJ

JJ 2

e ^ d e . , * = e ^ d e . . = a.. € &, ® P. Since P e . , €$ ®P , subtracting a suitable

JJ JJ JJ JJ JJ 1 JJ 0

scala r multiple of e.J.J from a.,J if necessary , we may assum e a = [ A

J i, * \

y

o^ .