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).
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