Journal of Lie Theory

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 {\bf P}$^{N-1}$. Suppose $n=4$, hence $N=24$. Take a random subspace of dimension $12$ in ${\bf P}^{23}$, 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 ${\bf{gl}}_2$, $195$ 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.