Bianchi's classification of 3-dimensional Lie algebras revisited

We present Bianchi's proof on the classification of real (and complex) $3$-dimensional Lie algebras in a coordinate free version from a strictly representation theoretic point of view. Nearby we also compute the automorphism groups and from this the orbit dimensions of the corresponding orbits in the algebraic variety $X\subseteq\Lambda^2V^*\otimes V$ describing all Lie brackets on a fixed vector space $V$ of dimension $3$. Moreover we clarify which orbits lie in the closure of a given orbit and therefore the topology on the orbit space $X/G$ with $G=\mathrm{Aut}(V)$.