Is $[I,J]$ a vector space where $I,J$ are ideals of a Lie algebra

Question

Let $\mathfrak g$ be a Lie algebra with ideals $I,J$. It is easy to check that $[[I,J],\mathfrak g] \subset [I,J]$ using the Jacobi identity but is it also a vector space?

That is, if $i,i' \in I$ and $j,j' \in J$, is it necessary that $[i,j] + [i',j'] \in [I,J]$?

Answer 1

Yes, $[I,J]+[I,J]\subset [I,J]$ by definition of $[I,J]$.

Answer 2

By definition $[I,J]$ is the vector space spanned by $[i,j]$ for $i\in I$ and $j\in J$, so yes.