I'm trying to find an example showing why, in a Lie algebra, we can't just define the product of two ideals $I$ and $J$ to be the elements of the form $[x,y]$ where $x \in I, \; y \in J$. I imagine such an example might be quite pathological, since none of the Lie algebras I'm more comfortable with seem to require this. I've tried working with matrices with entries in $\mathbb{C}[x,y]$, but haven't managed to get anywhere.
Would anyone be able to provide a link giving an example, or hints about how to construct such an example?