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Let $\Bbbk$ be an algebraically closed field of characteristic 0. We studysome cohomological properties of Lie subalgebras of the Witt algebra $W =\operatorname{Der}(\Bbbk[t,t^{-1}])$ and the one-sided Witt algebra $W_{\geq-1} = \operatorname{Der}(\Bbbk[t])$. In the first part of the paper, weconsider finite codimension subalgebras of $W_{\geq -1}$. We computederivations and one-dimensional extensions of such subalgebras. Thesecorrespond to $\operatorname{Ext}_{U(L)}^1(M,L)$, where $L$ is a subalgebra of$W_{\geq -1}$ and $M$ is a one-dimensional representation of $L$. We find thatthese subalgebras exhibit a kind of rigidity: their derivations and extensionsare controlled by the full one-sided Witt algebra. As an application of thesecomputations, we prove that any isomorphism between finite codimensionsubalgebras of $W_{\geq -1}$ extends to an automorphism of $W_{\geq -1}$.  The second part of the paper is devoted to explaining the observed rigidity.We define a notion of "completely non-split extension" and prove that $W_{\geq-1}$ is the universal completely non-split extension of any of its subalgebrasof finite codimension. In some sense, this means that even when studyingsubalgebras of $W_{\geq -1}$ as abstract Lie algebras, they remember that theyare contained in $W_{\geq -1}$. We also consider subalgebras of infinitecodimension, explaining the similarities and differences between the finite andinfinite codimension situations.  Almost all of the results above are also true for subalgebras of the Wittalgebra. We summarise results for $W$ at the end of the paper.

Derivations, extensions, and rigidity of subalgebras of the Witt algebra