  Lie algebras of dimension $n$ are defined by their structure constants ,
which can be seen as sets of $N = n^2 (n -- 1)/2$ scalars (if we take into
account the skew-symmetry condition) to which the Jacobi identity imposes
certain quadratic conditions. Up to rescaling, we can consider such a set as a
point in the projective space $P^{N--1}$. Suppose $n =4$, hence $N = 24$. Take
a random subspace of dimension $12$ in $P^{23}$ , over the complex numbers. We
prove that this subspace will contain exactly $1033$ points giving the
structure constants of some four dimensional Lie algebras. Among those, $660$
will be isomorphic to $gl\_2$ , $195$ will be the sum of two copies of the Lie
algebra of one dimensional affine transformations, $121$ will have an abelian,
three-dimensional derived algebra, and $57$ will have for derived algebra the
three dimensional Heisenberg algebra. This answers a question of Kirillov and
Neretin.
