This paper explicitly expresses all form of necessary and sufficient conditions for the decomposition of any kind of fourth-order linear time-varying system as a commutative pairs of third and first-order systems. Regarding the non-zero initial conditions, additional requirements are derived in order to satisfy the decomposition processed. Further, explicit formulas for reducing a fourth-order linear time-varying system into its third and first-order commutative pairs are derived and solved. The method points out the effect of disturbance and sensitivity on the systems. The results are illustrated by solving some examples.