  We give the complete classification of Mukai pairs of dimension $4$ and rank
$2$ with Picard number one, that is, pairs $(X,E)$ where $X$ is a Fano $4$-fold
with Picard number one, and $E$ is an ample vector bundle of rank two on $X$
with $c_1(X) = c_1(E)$. Equivalently, the present paper completes the
classification of ruled Fano $5$-folds with index two, which was partially done
by C. Novelli and G. Occhetta in 2007.
