  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 $G=\mathrm{Aut}(V)$.
