This paper presents a new simple algorithm that guarantees simultaneous conservation of energy, linear momentum, and angular momentum of a whole system in reaction dynamics calculations, employing atomic Cartesian coordinates. We apply this algorithm to the reaction dynamics in the NH3+ + NH3 system. We show that along the intrinsic reaction coordinate (IRC) of the hydrogen abstraction (HA) channel of the reaction, the geometries of local minima and transition state (TS) change appreciably with the rotational energy due to the angular momentum. Reaction dynamics calculations in the vicinity of the IRC reveal the dynamical effects that the angular momentum promotes or suppresses the reaction.