Let's say $L$ is an Lie algebra and $H$ is an subalgebra.
It's very common to talk about the ideal generated by a subalgebra, i.e., the intersection of all ideals containing a subalgebra. That's look like an analoguos of the normal closure in group theory.
But what about an analoguos for the normal core? How can I find an ideal of $L$ that is contained in $H$? And what about the largest ideal that satisfies it?
Edit: maybe I wasn't accurate enough when I asked my question. I was asking about an analoguous for the normal core in the context of Lie algebras, i.e., a maximal ideal of $L$ with respect to the property of being contained in $H$.
Some responses involved the solvable radical, but I think that's not a good option, because algebras like the free Lie algebra with rank 2 or greater are not solvable and nor are any of their ideals (if I got it right).
Best Answer
Take the sum of all ideals of $L$ that are contained in $H$. This will be the analogue of the normal core, i.e. the maximal ideal of $L$ contained in $H$.