Products of unipotent elements of index $2$ in orthogonal and symplectic groups

Abstract: An automorphism $u$ of a vector space is called unipotent of index $2$ whenever $(u-\mathrm{id})^2=0$. Let $b$ be a non-degenerate symmetric or skewsymmetric bilinear form on a vector space $V$ over a field $\mathbb{F}$ of characteristic different from $2$. Here, we characterize the elements of the isometry group of $b$ that are the product of two unipotent isometries of index $2$. In particular, if $b$ is symplectic we prove that an element of the symplectic group of $b$ is the product of two unipotent isometries of index $2$ if and only if it has no Jordan cell of odd size for the eigenvalue $-1$. As an application, we prove that every element of a symplectic group is the product of three unipotent elements of index $2$ (and no less in general). For orthogonal groups, the classification closely matches the classification of sums of two square-zero skewselfadjoint operators that was obtained in a recent article.