Let $G$ be a complex Lie group (not necessarily connected) with reductive Lie algebra $\frak{g}$. (We may assume that $G$ has finitely many connected components and is linear-algebraic.) Of course, $G$ need not be the complexification of a compact Lie group (ex. $G=\mathbb{C}$). To what extent, however, is $G$ "close" to being the complexification of a compact Lie group? Does $G$ belong to some kind of extension involving the complexification of a compact Lie group? Is $G$ some reasonably nice quotient of the complexification of a compact Lie group? I would appreciate any answers to questions of this nature. Also, I would appreciate any and all references.

Thanks!

reductive Lie groupin the header already raises questions about how you would define this notion. For linear algebraic groups the concept depends on the Jordan decomposition rather than the Lie algebra. Your 1-dimensional example shows the complication here, while your earlier question is relevant: mathoverflow.net/questions/124418 – Jim Humphreys Mar 20 '13 at 20:31commutativecompact complex Lie groups which have nothing to do with linear algebraic groups, namely the "complex tori" in the sense of $V/L$ for a finite-dimensional complex vector space $V$ and full rank lattice $L$ in $V$. So the structure of the center needs to be brought out in the analytic theory to "rule out" problematic cases. The description I gave with the Lie algebra and the center (and a bit for the component group) seems as "good" as one can hope to say over $\mathbf{C}$ (life is harder over $\mathbf{R}$), or maybe someone else has a better idea... – user28172 Mar 20 '13 at 23:46