  We study the problem of determining all connected Lie groups $G$ which have
the following property (hlp): every sub-Laplacian $L$ on $G$ is of holomorphic
$L^p$-type for $1\leq p<\infty, p\ne 2.$ First we show that semi-simple
non-compact Lie groups with finite center have this property. We then apply an
$L^p$-transference principle, essentially due to Anker, to show that every
connected Lie group $G$ whose semi-simple quotient by its radical is
non-compact has property (hlp). For the convenience of the reader, we give a
self-contained proof of this transference principle, which generalizes the
well-known Coifman-Weiss principle. One is thus reduced to studying compact
extensions of solvable Lie groups. We extend previous work of Hebisch, Ludwig
and M\"uller to compact extensions of certain classes of exponential solvable
Lie groups.
