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).