  The main aim of this work is to construct several new families of proper
biharmonic functions defined on open subsets of the classical compact simple
Lie groups $\SU n$, $\SO n$ and $\Sp n$. We work in a geometric setting which
connects our study with the theory of submersive harmonic morphisms. We develop
a general duality principle and use this to interpret our new examples on the
Euclidean sphere $\s ^3$ and on the hyperbolic space $\H^3$.
