In this paper we prove that there are infinitely many abelian left symmetric algebras in dimensions
≥
6
\geq 6
. Equivalently this means that there are, up to affine conjugation, infinitely many simply transitive affine actions of
R
k
\mathbb R^k
, for
k
≥
6
k\geq 6
. This is a result which is usually credited to A.T. Vasquez, but for which there is no proof in the literature.