Producing non-abelian Lie algebra from abelian one
Let $\mathfrak{g}$ be a finite-dimensional abelian Lie algebra over a field $k$ of characteristic zero. I was wondering how many constructions do we have to produce a non-abelian Lie algebra $\overline{\mathfrak{g}}$ out of $\mathfrak{g}$ i.e. using just Lie algebra $\mathfrak{g}$ itself, no central extension/semi-direct product.
So far, I have just two constructions:
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Consider the Lie algebra of derivations $Der(\mathfrak{g})$. In this case, we are going to get the set of linear maps on $\mathfrak{g}$.
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Consider $\overline{\mathfrak{g}}=\mathfrak{g}\oplus\wedge^2\mathfrak{g}$ with a bracket $[x,y]=x\wedge y$ if $x,y\in \mathfrak{g}$ and $[x,y\wedge z]=[x\wedge y,z]=[x\wedge y, z\wedge w]=0$. Is there something specific about this particular Lie algebra?
Are there any other way to produce a non-abelian Lie algebra?
As other people have said, you might as well say how can you build Lie algebras from a vector space $V$. There are bewilderingly many in general. If we restrict our attention to semisimple ones then we can easily construct $\mathfrak{sl}(V)$ as the tracefree linear maps and $\mathfrak{so}(V)$, $\mathfrak{sp}(V)$ (the second only if $V$ is even dimensional) as subspaces of $\mathfrak{sl}(V)$ which are skew for an appropriate symmetric or symplectic form on $V$. Note these two are isomorphic to $\bigwedge^2 V $, $S^2 V$.
We can also make many more by doing the same to $V \oplus V$, $V \otimes V$ and so on, or by sums and products of our resulting Lie algebras. You can do tonnes of different things here, especially if we're allowed to split $V$ up etc. If $V$ is 1-dimensional we can reconstruct the entirety of the classification of semisimple Lie algebras in this way.