  Any two-qubit state can be represented geometrically by a steering ellipsoid
inside the Bloch sphere. We extend this approach to represent any block
positive two-qubit operator $B$. We derive a novel classification scheme based
on the positivity of $\det B$ and $\det B^\mathrm{T_B}$; this shows that any
ellipsoid inside the Bloch sphere must represent either a two-qubit state or a
two-qubit entanglement witness. We focus on such witnesses and their
corresponding ellipsoids, finding that properties such as witness optimality
are naturally manifest in this geometric representation.
