Ideals in Lie algebras

Question

Is it true that in a Lie algebra $\mathcal {L}$ the product of two ideals $[I, J]$ is equal to the intersection $ I\cap J $?

Answer 1

No. Take $L$ to be abelian and $I, J$ to be two subspaces of $L$ (which are automatically ideals). $[I, J]$ is always zero, but $I \cap J$ need not be. In general you only have an inclusion $[I, J] \subseteq I \cap J$.

Answer 2

We know that always $\left[I,J\right]\subseteq I\cap J$; however, the converse is false.

For example, in $L=\text{gl}\left(n,F\right)$ (over an arbitrary field $F$), take $I=L$ and $J=Z\left(L\right)=\left\{\text{scalar matrices}\right\}$. Then $\left[I,J\right]=0$, whereas $I\cap J=J$.