This is a homework question, in which we've got a bunch of kinds of subsets of a given Lie-algebra, and needed to decide wether these are sub-algebras, ideals, or non of the above. I have managed to find all besides 2 counter examples:
Given a commutative ring $k$ and $k$-Lie-algebra $A$, and $I,J\subset A$ are sub-algebras. then define: $\left[I,J\right]=\mbox{span}\left\{ \left[a,b\right]\,|\, a\in I,\, b\in J\right\}$ . this is obviously a $k$-modul. I'm pretty sure that $\left[I,J\right]$ is not necessarily a sub-algebra, but i can't find a counter example.
The other case, is exactly like the above, only this time $I$ is an ideal ($J$ is not an necessarily an ideal). this time i proved that this is a sub-algebra, but i'm pretty sure it is not necessarily an ideal, yet niether for this could i find a counter-example.
Any ideas for those 2?
Thanks alot!