I'm new to Lie algebras.
Let $\mathfrak{g}$ be a Lie Algebra with the bracket $[a,b]=ab-ba$.
and $I$ and $J$ are two Ideals, which means that:
$$[I,\mathfrak{g}]\subset I$$ and $$[J,\mathfrak{g}]\subset J.$$
So I have already proved that the subset $I+J$ is also an ideal,
but I have a problem when I tried to prove that the subset $IJ$ defined by
$$ IJ=\{z \in \mathfrak{g} \quad |\quad \exists (x,y)\in I\times J, \; z=xy\}$$
is also an ideal.
I tried this :
we need to prove that
$$ \forall y \in\mathfrak{g},\quad \forall x\in IJ \quad [x,y] \in IJ $$
let $x$ be in $IJ$ then $\exists (a,b)\in I\times J$ with $x=ab$
$$ [x,y]=[ab,y]=aby-yab=...=a[b,y]+[a,y]b$$
we know that $[b,y] \in J $ because $J$ is ideal then: $$a[b,y]\in IJ$$ by the same way $$[a,y]b\in IJ$$ My problem is that I can NOT prove that $a[b,y]+[a,y]b \in IJ$ cause nothing says that "$+$" is inner map in $IJ$.