We will study the norm-closure of the set $\mathfrak{C}_{\mathfrak{E}}$ of bounded linear operators acting on a complex, separable Hilbert space $\mathcal{H}$ which may be expressed as the commutator of two idempotent operators. In particular, we will identify which biquasitriangular operators belong to the norm-closure of $\mathfrak{C}_{\mathfrak{E}}$.

If time permitted, we will also give characterizations of matrices could be expressed as the commutator of two square zero matrices, and some related results about limits of commutators of two square zero operators acting on $\mathcal{H}$.

This is based on joint papers with Laurent Marcoux and Heydar Radjavi.