  Let $\mathcal{H}$ be a complex, separable Hilbert space, and set
$\mathfrak{c}($NIL$_2)=\{ MN - NM : N, M \in \mathcal{B}(\mathcal{H}), M^2 = 0
= N^2 \}$. When $\dim\, \mathcal{H}$ is finite, we characterise the set
$\mathfrak{c}($NIL$_2)$ and its norm-closure CLOS$(\mathfrak{c}($NIL$_2))$. In
the infinite-dimensional setting, we characterise the intersection of
CLOS$(\mathfrak{c}($NIL$_2))$ with the set of biquasitriangular operators, and
we exhibit an index obstruction to belonging to CLOS$(\mathfrak{c}($NIL$_2))$.
