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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.

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I think I would just say "the ideal generated by $\mathfrak{m}$". – Tom De Medts Mar 13 '13 at 12:39
Probably there is no special terminology, since the classical structure theory didn't make use of this procedure. What Tom suggests is reasonable enough. (But if you want to look further into the literature, try the papers by D.A. Towers. There is a British tradition of studying abstract Lie algebras in the spirit of infinite group theory.) – Jim Humphreys Mar 13 '13 at 13:43
up vote 3 down vote accepted

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.

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