Abstract: We classify those finite dimensional Lie algebras, over a field $\k$ of characteristic zero, whose cohomology with trivial coefficients has dimension 2. We show that the only such algebras are the 3-dimensional simple algebras and the semi-direct products $\n\rtimes_\phi \k$, where $\n$ is a nilpotent Lie algebra and $\phi\colon\n\to \n$ is a derivation which induces a non-singular map in each cohomology space $H^i(\n)$, for $i>0$.
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