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In my opinion, it's probably not so interesting to consider all extensions. If you consider the Lie subalgebra $\mathfrak{n}_n\subset \mathfrak{gl}_n$ of all strictly upper triangular matrices, then $\mathfrak{n}_n$ is an extension of $\mathbf{C}$ because it has a quotient Lie algebra of dimension 1. It follows that even for nilpotent extensions the nilpotency can grow arbitrarily large. Of course extensions can be even more chaotic than that.