On some simple groups defined by C. Chevalley

RIMHAK REEIntroduction.For any semi-simple Lie algebra g over the field of complex numbers and an arbitrary field K, C. Chevalley [2] defined a group © by a uniform method.The group © turns out to be simple when g is simple (except for a few exceptional cases), and yields some new classes of simple groups not contained in the theory of classical groups, if, in particular, g is one of the exceptional simple Lie algebras.In this paper first we consider the case where g is one of the classical algebras An, Bn, C", Dn, and identify the group © with classical groups, thus answering a question raised in the last section of [2].Our results are as follows:(a) if g is of the type An, then © is the special projective group PSL(»+1, K);(b) if g is of the type Cn, then © is the quotient group of the symplectic group Sp(2ra, K) over its center;(c) if g is of the type £>", then © is the commutator group of the projective orthogonal group defined by the quadratic form 5^**=J £*£-* (3 gra);(d) if g is of the type Bn and if K is not of characteristic 2 then © is the commutator group of the projective orthogonal group defined by the quadratic form yiLn £*£_* (2 gra); if g is of type Bn and if K is of characteristic 2 then © is a subgroup of Sp (2ra, K); if, moreover, K is perfect(x) then G = Sp(2ra, K).The author has been unable to identify © in case K is not perfect and of characteristic 2. Secondly, we consider the case where g is the exceptional algebra G2, and