14
M . L. RACINE
LEMMA 1. Let #/K be a Jordan algebra M an order of 9. The
following are equivalent.
(i) M is a maximal order of #.
(ii) M is a maximal order of 9 f ° r a H P S.
(iii) M is a maximal order of 5 for all p e S.
§3. Maximal orders of Jordan algebras of degree 2.
Let ?/K be a central simple Jordan algebra of degree 2. Then
9 = P(N, 1) the quadratic Jordan algebra of the quadratic form N with bas e
point 1, dim $ _ 3 and N is a non-degenerate quadratic form. We refer to
K
[19] for result s about thes e algebras .
PROPOSITION 4. Automorphisms of £ = «?(N, 1) coincide with
isometries of the quadratic form N which fix 1.
PROOF. The result follows immediately from (11) - (13).
PROPOSITION 5. Let M be an o-lattice of # = £(N, 1) containing 1.
The following are equivalent.
(i) N is integral on M.
(ii) All elements of M are integral.
(iii) M is an order of p.
2
(iv) x

M implies x
e
M.
PROOF. Assume (i); since T(x) = N ( l , x )

o for x e M, (i) = (ii).
Assume (ii). Let x, y

M. Then T(y)
6
o and y = T(y)l - y

M. Therefore
N(x,y)

o, N(x)
e
o and yU = N(xty)x - N(x) y

M. Hence (ii) = (iii).
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