ARITHMETICS OF JORDAN ALGEBRAS 3 7

obtained in this way ([17], p. 0. 7). Assume A* = HA H = A. Let

B = H"1A; I * = A^H" 1 = H^AHH"" 1 = H - 1 A if H is hermitian, -H" 1 A if

H is skew-hermitian. Therefore A = HB, B hermitian or skew-hermitian

according a s H is hermitian or skew hermitian.

THEOREM 1. Let * be the involution of & determined by h. An

n

order F of & is maximal ^-stabl e if and only if there exist an O-lattice L

n

of V such that

(i) F = E(L) n E(L)*.

(ii) Any elementary divisor of H the matrix of h with respect

to {x,} a bas e of L (i.e . L = Ox 0 . . . 0 Ox ) is 1 or p.

PROOF. The order F is maximal ^-stabl e if and only if

F = E(L) nE(L)# for some lattice L and the expression is unique. Let

r l r n

p , . . . , p be the elementary divisors of H the matrix of h with respect

to a bas e of L. With respect to that bas e E = O and E* = HO H~ SO

n n

F = O nHO H . By Corollary 2 the expression is unique if and only if

r - I-J 1. If r = 2r, L = p~rL, E(L ) = E(L) and the matrix of h re L

has elementary divisors 1 or p. If r = 2r - 1 the matrix H of h re

Lj = p" r L has elementary divisors 1 or p " , E(L )* = H r» H*"1 = E(L H~ 2 ) .

The matrix of h re I ^ H " 1 is (hfx.H" 1 , X . H " 1 ) ) = H ^ H H " U = H " 1 and has

elementary divisors 1 or p. Hence E = E(L H )nE( L H )* ha s the

desired properties.

q . e . d .

This transformation L -* LH will play an important role in the next chapter