EXTENSIONS OF THE JACOBI IDENTITY 19

exists by extracting, for instance, the coefficient of a monomial in z2, z21, zA2

and z32. This implies the existence of (1.49) (which can also be seen directly)

by means of the "extended" Jacobi identity (1.23). Then we see that

Y(Y(Y(v4,z42)Y(v3,z32)v2,z21)v1,z1) ( l ^ 1 * ( £ ^ S l ) J •

exists by means of the coefficient of z\3z32z31, i,j £ Z. Since (1.50) is the

sum of two such terms, it too exists, and, applying the Jacobi identity, so

does (1.51). The existence of (1.52) is then a consequence of the identities

between the previous expressions. Since it will be relevant in the general

case, we show direct proof of the existence of (1.51) based on Lemma 1.6.

We use Lemma 1.6 to rewrite (1.51) as

Y(Y(Y(Y(v4,z43)v3,z32)v2,z21)v1,z1) J ] * * - i * f^*"'****"*)

C1-5?)

(expanded as in Lemma 1.6). The coefficient of

ltj4

in (1.57) equals

Y(Y(Y(Y(vAlz43)v39zS2)v29z21)vuz1) U

[ 2 M - I )

• (1-58)

iti4 \*=t /

Expanding ^(^4,^43) according to (1.4), and using (1.5), we can write the

coefficient of z[3, I G Z, in (1.58) as a finite linear combination of expressions

of the form

Y(Y(Y(w,z32)v2jz21)vuz1)(z1 + z2l)i(zl + z21 + ^32)^^21 + z32)kz%2, (1.59)

where i, j , k and n are fixed integers and w £ V. The existence if (1.41) (cf.

Lemma 1.3) shows the existence of (1.59). •