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^ .
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