348 В. van Geemen

Proof If F is a relation of type G^ the statement follows from Proposition (4.15), where we take F = (*) and (**) is then g(F). If F is of type Gß then F can be written as :

^\ / / ' [ o !']^'^^'[l e']^'^ = ^ forallxeM, Hence, by Proposition (4.15), the polynomial: ZA^CXC-ir^^^'oMoCa + lO 6)])(X(-ir^'''0M0[cT + (O £)]) is zero, hence the polynomial

ь^и ( ^ - ' - - С IK 'T])

is also zero. Using formula (3) we find (for all lelHg^i):

. ^4 V-. nfO 0 el Г0 0 el rO 0 e 1 , ГО 0 el ^

( note that we found in fact a conjugate of cr(F)).

( 4 . 19 ) Finally we describe the known quartic relations between the Ö I. The monomials occuring in such a relation are: ^^ -*

where (/ч)» (n,)» ( ,)' (i/ J is an isotropic subspace V of the symplectic

vector space ^2^^, i.e. '^иД' + */г'Д = 0. Using the action of JQ on the Ö I we find: Le J

( 4 . 20 ) Lemma. A quartic relation of type (4.19) is the sum of a quadratic relation and a sum of relations which are conjugates under the action of F^ of a relation of type:

. Ч V-. nfO 0 elnfO 0 el^rO 0 eirO 0 el

Proof Similar to the proof of the Lemmas 1 and 2.

The following proposition describes the relations of type (***). Its proof is similar to the proof of Proposition (4.15) and we omit it. (Note that part of it is in fact given in the proof of Proposition (4.18).)