On some 4-by-4 matrices with bi-elliptical numerical ranges

A complete description of 4-by-4 matrices $\begin{bmatrix}\alpha I & C \\D & \beta I\end{bmatrix}$, with scalar 2-by-2 diagonal blocks, for which the numerical range is the convex hull of two non-concentric ellipses is given. This result is obtained by reduction to the leading special case in which $C-D^*$ also is a scalar multiple of the identity. In particular cases when in addition $\alpha-\beta$ is real or pure imaginary, the results take an especially simple form. An application to reciprocal matrices is provided.