  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.
