I have some elementary questions about Lie algebras and vector space complements.

Let $(\mathfrak{g},[.,.])$ be a finite-dimensional Lie algebra and $\mathfrak{g}_1$ a Lie ideal in $\mathfrak{g}$.

1) Is it possible to choose $\mathfrak{g}$ and $\mathfrak{g}_1$ in such a way that there is no vector space complement $\mathfrak{g}_2$ of $\mathfrak{g}_1$ in $\mathfrak{g}$ which is additionally a sub-Lie algebra of $\mathfrak{g}$?

2) Is there an example of $\mathfrak{g}$ and $\mathfrak{g}_1$ such that there is a vector space complement $\mathfrak{g}_2$ of $\mathfrak{g}_1$ in $\mathfrak{g}$ with $[\xi_1,\xi_2] = 0$ for all $\xi_1 \in \mathfrak{g}_1$ and $\xi_2 \in \mathfrak{g}_2$ but such that no such complement is a Lie ideal in $\mathfrak{g}$?

In the situation I need, all Lie algebras occur as Lie algebras of some Lie group. For every Lie algebra $\mathfrak{g}$ there exists a Lie group $G$ with $\mathrm{Lie}(G) = \mathfrak{g}$, but that's rather abstract, so it would be nice if $\mathfrak{g}$ and $\mathfrak{g}_1$ were the Lie algebras of some well-known Lie groups.

I'm not an expert in Lie algebra theory, so if some things described in 1) or 2) have special names in the literature it would be nice if you could mention that or give some references.

reductive, I have found that in O'Neill, Semi-Riemannian-Geometry and Baums "Eichfeldtheorie" book. But I havn't found any more comprehensive references. $\endgroup$ – student Jul 16 '10 at 13:42