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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 orskewsymmetric bilinear form on a vector space $V$ over a field $\mathbb{F}$ ofcharacteristic different from $2$.  Here, we characterize the elements of the isometry group of $b$ that are theproduct of two unipotent isometries of index $2$. In particular, if $b$ issymplectic we prove that an element of the symplectic group of $b$ is theproduct of two unipotent isometries of index $2$ if and only if it has noJordan cell of odd size for the eigenvalue $-1$. As an application, we provethat every element of a symplectic group is the product of three unipotentelements of index $2$ (and no less in general).  For orthogonal groups, the classification closely matches the classificationof sums of two square-zero skewselfadjoint operators that was obtained in arecent article.

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