  By a Sugawara construction we mean a generalized Virasoro construction in
which the currents are primary fields of conformal weight one. For simple Lie
algebras, this singles out the standard Sugawara construction out of all the
solutions to the Virasoro master equation. Examples of nonsemisimple Sugawara
constructions have appeared recently. They share the properties that the
Virasoro central charge is an integer equal to the dimension of the Lie algebra
and that they can be obtained by high-level contraction of reductive Sugawara
constructions: they thus correspond to free bosons. Exploiting a recent
structure theorem for Lie algebras with an invariant metric, we are able to
unify all the known constructions under the same formalism and, at the same
time, to prove several results about the Sugawara constructions. In particular,
we prove that all such constructions factorize into a standard (semisimple)
Sugawara construction and a nonsemisimple one (with integral central charge) of
a form which generalizes the nonsemisimple examples known so far.
