  We introduce a new method for constructing complex-valued $r$-harmonic
functions on Riemannian manifolds. We then apply this method for the important
semisimple Lie groups $SO(n)$, $SU(n)$, $Sp(n)$, $SL_n(R)$, $Sp(R,n)$,
$SU(p,q)$, $SO(p,q)$, $Sp(p,q)$, $SO^*(2n)$ and $SU^*(2n)$.
