Let $\mathfrak{m}$ be a Lie sub-algebra 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.