We review the Thomas precession exhibiting the exact form of the Thomas rotation in the axis-angle parameterization. Assuming three inertial frames $S, S', S''$ moving with arbitrary velocities and with $S, S''$ having their axis parallel to the axis of $S'$ we focus our attention on the two essential elements of the Thomas precession e.g., (i) there is a rotation between the axis of frames $S$, $S''$ and (ii) the combination of two Lorentz transformations from $S$ to $S'$ and from $S'$ to $S''$ fails to produce a pure Lorentz transformation from $S$ to $S''$. The physical consequence of (i) and (ii) refers to the impossibility of having arbitrary frames $S, S', S''$ moving with their axis mutually parallel. Then, we reexamine the validity of (i) and (ii) under the conjecture the time depends on the state of motion of the frames and we show that the Thomas precession assumes a different form as formulated in (i) and (ii).