16

CRISTIANO HUSU

[Y(v4,z4) xZil [Y{v3,z3) xZ31 \Y{v2,z2) xZ21 y(vj,*i)]]] =

= Y(Y(v4,z41)Y(v3,z31)Y(v2,z2l)vl,z1) f[ z^6 (2—&\ (1.47)

and similar identities (with the same product of ^-functions) obtained by

permuting 2,3,4. The next step is to consider

Y([Y(v4,z41) xZi2 [Y(v3,z31) xZ32 YfaznW^zJ f [ z^8 ( 5 i — 2

(1.48)

which, again using (1.23), equals

Y(Y(Y(v4,z42)Y(v3,z32)v2lz2l)v1,zl) l j [

z

- ^ ( ^ ^ j J •

Similar expressions are obtained permuting the indices 3 and 4 in (1.48) and

(1.49) (again with the same products of ^-functions). Then, in the same

spirit, we form

Y(Y([Y(v4,z42)xZi3Y(v3,z32)]v2,z2l)vuZl)l[ f[ z^S f ^

1

"

ZA

,

(1.50)

where we set

^to

=

2i,

1 i 4.

Applying the Jacobi identity, on the one hand, we write (1.50) as

Y(Y(Y(Y(v4,z43)v3,z32)v2,z21)v1,z1) JJ 'v-i* fr'*'1 ~ **) , (1-51)

ltj4 V *t,i-l /

while, on the other hand, using (1.47) — (1-49), we see that (1.50) equals