  In the context of weak KAM theory, we discuss the commutators $\{T^-_t\circ
T^+_t\}_{t\geqslant0}$ and $\{T^+_t\circ T^-_t\}_{t\geqslant0}$ of Lax-Oleinik
operators. We characterize the relation $T^-_t\circ T^+_t=Id$ for both small
time and arbitrary time $t$. We show this relation characterizes
controllability for evolutionary Hamilton-Jacobi equation. Based on our
previous work on the cut locus of viscosity solution, we refine our analysis of
the cut time function $\tau$ in terms of commutators $T^+_t\circ
T^-_t-T^+_t\circ T^-_t$ and clarify the structure of the super/sub-level set of
the cut time function $\tau$.
