  Let $\mathcal{G}_k$ be the gauge group of the principal $SU(4)$-bundle over
$S^4$ with second Chern class $k$ and let $p$ be a prime. We show that there is
a rational or $p$-local homotopy equivalence
$\Omega\mathcal{G}_k\simeq\Omega\mathcal{G}_{k'}$ if and only if
$(60,k)=(60,k')$.
