  Holomorphic functions of exponential type on a complex Lie group $G$
(introduced by Akbarov) form a locally convex algebra, which is denoted by
$\cO_{exp}(G)$. Our aim is to describe the structure of $\cO_{exp}(G)$ in the
case when $G$ is connected. The following topics are auxiliary for the claimed
purpose but of independent interest: (1) a characterization of linear complex
Lie group (a~result similar to that of Luminet and Valette for real Lie
groups); (2) properties of the exponential radical when $G$ is linear; (3) an
asymptotic decomposition of a word length function into a sum of three summands
(again for linear groups). The main result presents $\cO_{exp}(G)$ as a
complete projective tensor of three factors, corresponding to the length
function decomposition. As an application, it is shown that if $G$ is linear
then the Arens-Michael envelope of $\cO_{exp}(G)$ is just the algebra of all
holomorphic functions.
