We present a new proof of the identities needed to exhibit an explicit Z-basis for
the universal enveloping algebra associated to an affine Lie algebra. We then use the
explicit Z-bases to extend Borcherds' description, via vertex operator representations,
of a Z-form of the enveloping algebras for the simply-laced affine Lie algebras to the
enveloping algebras associated to the unequal root length affine Lie algebras.
Keywords and phrases. Affine Lie algebra, integral basis, universal enveloping algebra,
vertex operator algebra.
Received by the editor August 27, 1990.
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