In the answer of my question:


[On the full reducibility of representations of reductive Lie algebras](https://mathoverflow.net/questions/23085/on-the-full-reducibility-of-representations-of-reductive-lie-algebras)


James E. Humphreys replied to me saying that:"the notion of "reductive" for a Lie algebra in characteristic 0 has no intrinsic interest, unless you study the Lie algebra of a Lie (or algebraic) group and relate their representations carefully."


Can please someone explain that to me or give to me any reference?
thank you!



Reductive Lie algebra of a Lie group
What Jim means is that one naive definition of reductive Lie algebra 


* $\mathfrak{g}$ is **reductive** if all its finite-dimensional representations are semi-simple.


already has a name: **semi-simple.**


Another one


* $\mathfrak{g}$ is **reductive** if all its representations are semi-simple.


is actually trivial; there are no (**EDIT:** nonzero) Lie algebras that satisfy it.


Of course, there actually is a pretty good definition that matches better with reductive for groups:


* $\mathfrak{g}$ is **reductive** if its adjoint representation is semi-simple.


but it's important to keep in mind that the properties above don't follow from that.


