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
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