Perfect lattice (awaiting review)

A lattice $L$ in $\mathbb R^n$ is called perfect if the space of $n \times n$ real symmetric matrices is spanned by the set of $n \times n$ symmetric matrices $$\{ x x^{\top} : x \in S(L) \},$$ where $S(L)$ is the set of minimal vectors of $L$. A perfect lattice is well-rounded, arithmetic, and is uniquely determined by its minimal vectors. Further, there are only finitely many similarity classes of perfect lattices in each dimension $n \geq 2$.