  Let $G$ be a connected exponential Lie group and $R$ be the solvable radical
of $G$. We describe a condition on $G/R$ under which one can then conclude that
$R$ is an exponential Lie group. The condition holds in particular when $G$ is
a complex Lie group and this yields a stronger version of a result of Moskowitz
and Sacksteder \cite{MS} on the center of a complex exponential Lie group being
connected. Along the way we prove a criterion for elements from certain subsets
of a solvable Lie group to be exponential, which would be of independent
interest.
