  Let $b$ be a non-degenerate symmetric (respectively, alternating) bilinear
form on a finite-dimensional vector space $V$, over a field with characteristic
different from $2$. In a previous work, we have determined the maximal possible
dimension for a linear subspace of $b$-alternating (respectively,
$b$-symmetric) nilpotent endomorphisms of $V$. Here, provided that the
cardinality of the underlying field be large enough with respect to the Witt
index of $b$, we classify the spaces that have the maximal possible dimension.
  Our proof is based on a new sufficient condition for the reducibility of a
vector space of nilpotent linear operators. To illustrate the power of that new
technique, we use it to give a short new proof of the classical Gerstenhaber
theorem on large vector spaces of nilpotent matrices (provided, again, that the
cardinality of the underlying field be large enough).
