On commutators of square-zero Hilbert space operators

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${}_2))$.