Context. Stellar rotation affects the transport of chemical elements and angular momentum and is therefore a key process during stellar evolution, which is still not fully understood. This is especially true for massive OB-type stars, which are important for the chemical enrichment of the Universe. It is therefore important to constrain the physical parameters and internal angular momentum distribution of massive OB-type stars to calibrate stellar structure and evolution models. Stellar internal rotation can be probed through asteroseismic studies of rotationally split non radial oscillations but such results are still quite rare, especially for stars more massive than the Sun. The slowly pulsating B9V star HD 201433 is known to be part of a single-lined spectroscopic triple system, with two low-mass companions orbiting with periods of about 3.3 and 154 days. Aims: Our goal is to measure the internal rotation profile of HD 201433 and investigate the tidal interaction with the close companion. Methods: We used probabilistic methods to analyse the BRITE - Constellation photometry and radial velocity measurements, to identify a representative stellar model, and to determine the internal rotation profile of the star. Results: Our results are based on photometric observations made by BRITE - Constellation and the Solar Mass Ejection Imager on board the Coriolis satellite, high-resolution spectroscopy, and more than 96 yr of radial velocity measurements. We identify a sequence of nine frequency doublets in the photometric time series, consistent with rotationally split dipole modes with a period spacing of about 5030 s. We establish that HD 201433 is in principle a solid-body rotator with a very slow rotation period of 297 ± 76 days. Tidal interaction with the inner companion has, however, significantly accelerated the spin of the surface layers by a factor of approximately one hundred. The angular momentum transfer onto the surface of HD 201433 is also reflected by the statistically significant decrease of the orbital period of about 0.9 s during the last 96 yr. Conclusions: Combining the asteroseismic inferences with the spectroscopic measurements and the orbital analysis of the inner binary system, we conclude that tidal interactions between the central SPB star and its inner companion have almost circularised the orbit. They have, however, not yet aligned all spins of the system and have just begun to synchronise rotation. Based on data collected by the BRITE - Constellation satellite mission, built, launched and operated thanks to support from the Austrian Aeronautics and Space Agency and the University of Vienna, the Canadian Space Agency (CSA), and the Foundation for Polish Science & Technology (FNiTP MNiSW) and National Science Centre (NCN), the Hermes spectrograph mounted on the 1.2 m Mercator Telescope at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias, and the Solar Mass Ejection Imager, which is a joint project of the University of California San Diego, Boston College, the University of Birmingham (UK), and the Air Force Research Laboratory.