  In this paper we introduce a new method for manufacturing harmonic morphisms
from semi-Riemannian manifolds. This is employed to yield a variety of new
examples from the compact Lie groups SO(n), SU(n) and Sp(n) equipped with their
standard Riemannian metrics. We develop a duality principle and show how this
can be used to construct the first known examples of harmonic morphisms from
the non-compact Lie groups SL(n,R), SU(2n), Sp(n,R), SO(2n), SO(p,q), SU(p,q)
and Sp(p,q) equipped with their standard dual semi-Riemannian metrics.
