  In this paper, we analyze the possible homotopy types of the total space of a
principal $SU(2)$-bundle over a $3$-connected $8$-dimensional Poincar\'{e}
duality complex. Along the way, we also classify the $3$-connected
$11$-dimensional complexes $E$ formed from a wedge of $S^4$ and $S^7$ by
attaching a $11$-cell.
