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Summary: Reproduced (with corrections) from
The Punjab University
Journal of Mathematics
Vol. XVI (1983), pp. 930.
THE JACOBI IDENTITY
H. Azad
Introduction
The aim of this paper is to outline an alternative approach to Chevalley groups which
is suggested by results of R. Steinberg, especially § 11 of [6], and by [1]. The approach
we have in mind works with a system of axioms which involve only a root system and
a commutative ring, and in a sense avoids Chevalley bases. Needless to say, this would
have been impossible without knowing the contents of [2] and [6]. An advantage of
this approach is that problems like those mentioned in [2, p. 64] vanish automatically.
This paper is organized as follows: In § 1 we prove an analogue of [1] for a class of Lie
algebras. Then, in § 2, by simply reversing a procedure given in the proof of Proposition
(1.1), we construct, for a given root system which has no multiple bonds, a function N,
defined on pairs of independent roots (u, v) such that Nu,v is ±1 if and only if u+v is a
root, and verify the Jacobi identity for N. That such a function exists is nothing new;
see, for example [2, p. 24], [8] or [5, p. 285], which also gives the briefest solution to
date of this problem. We have thought doing this worthwhile as the function N arises
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