  We prove that the loop space of a quasitoric manifold is homotopy commutative
if and only if the underlying polytope is $(\Delta^3)^n$ and the characteristic
matrix is equivalent to a matrix of certain type. We also construct for each
$n\ge 2$ and a positive integer $k$, a quasitoric manifold $M(k,n)$ over
$(\Delta^3)^n$ such that its loop space is homotopy commutative if and only if
$k$ is even, where every quasitoric manifold over $\Delta^3$ is equivalent to
$\mathbb{C} P^3$ whose loop space is homotopy commutative.
