Let $\mathfrak{m}$ be a Lie subalgebra of the Lie algebra $\mathfrak{g}$. Is there a name for the smallest ideal of $\mathfrak{g}$ containing $\mathfrak{m}$? It certainly exists and coincides with the intersection of all the ideals containing $\mathfrak{m}$. The analogy with normal closure in group theory would suggest "ideal closure", but I don't remember having seen this terminology before.

In analogy with the notion of normal closure in group theory, the smallest ideal of ${\mathfrak g}$ containing the subalgebra ${\mathfrak m}$ is indeed called the ideal closure of ${\mathfrak m}$ and denoted by ${\mathfrak m}^{\mathfrak g}$. This terminology is rather diffused in the literature: for instance, you can find it in many papers by Amayo, Stewart, Towers, etc. 

