Let $L$ be a finite dimensional Lie algebra. I am getting rather confused with the notation/definition and I would appreciate any clarification.
1) When one talks about the commutator subalgebra (or the derived algebra) the notation used is $[L,L]$. And my understanding is that this means the set of all linear combinations of terms of the form $[x,y]$ with $x,y \in L$.
2) A subspace $H \subseteq L$ that is closed under the Lie bracket is called a Lie subalgebra. So that means $[h_1, h_2] \in H$ for all $h_1, h_2 \in H$?
3) If a subspace $I \subseteq L$ satisfies a stronger condition that $[L,I] \subseteq L$, then $I$ is called an ideal in the Lie algebra $L$. Does this mean $[x, i] \in I$ for all $x\in L, i \in I$? (or does it mean all lienar combinations of the terms $[x,i]$ is contained in $I$? I guess I'm wondering what $[L,I]$ is supposed to mean here)
Thank you very much.
Since you're wondering what $[L,I]$ is supposed to mean here, let me say that for subalgebras $I,J$ of $L$ the subspace $[I,J]$ is defined to be the linear span of all $[i,j]$ with $i\in I$ and $j\in J$. Note that $$ [I,J]=[J,I] $$ since the Lie bracket is anti-commutative and $I=-I$ for any subspace.