  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.
