ARITHMETICS OF JORDAN ALGEBRAS 43

isotope of M(& , & ). The triple (&, $ , -) is sometimes referred to as a

coordinate algebra. Since the norms of & are contained in $ , our previous

argument shows that v(& ) = 22 or 22£. Altering our previous definition

slightly (in characteristic 2) we will say that fl has a symmetric prime if

v(# ) = S. In that case pick p

€

&Q.

Let ^ = M(& , & , *). To determine maximal orders of # it suffices

to determine which maximal ^-stable orders E(L)

nE(L)";

induce a maximal

order M = ^ n (E(L) n E(L)*) of ^. By Lemma 13 L may be written as a

direct sum of mutually orthogonal lines and planes,

L ~

(cy

x . . .

x(dr)

x

I**1

°

r+ 1

] x

.. .

x

Z^-

1

^"M

, d.

e

*0.'

c

0 " J

6

C

.i

d

,o /

\

C

i

d

r+1 r+2' \ n-1 n

Clearly d., 1 i r is a unit of © times an elementary divisor of H the

1 /dP

° \

matrix of h with respect to a base of L. Now/ ^ ^ (satisfies

. ;

I d J » I ci^ | | c | (otherwise the proof of Lemma 13 shows that it can be

written as the sum of two lines) and

The first two matrices are units of & and | c - d c d- | = | c j . Hence

^ XJ XJ XJ

i/T±

XJ

the elementary divisors of [ j are p , p and the elementary

C

i

d

i+ l

divisors of H are determined by the c 's (corresponding to subnormal