We compare with one-dimensional particle-in-cell simulations the aperiodically growing instabilities driven by a bi-Maxwellian velocity distribution in unmagnetized electron plasma (Weibel instability) and in pair plasma. The simulation box is aligned with the cool direction. The waves in both simulations evolve towards a circularly polarized non-propagating magnetic structure. Its current and magnetic field are aligned and the structure is in a force-free state. We examine how a background magnetic field B₀, which is parallel to the simulation direction, affects the waves in the pair plasma. A weak B₀ cannot inhibit the growth of the aperiodically growing instability but it prevents it from reaching the force-free stable state. The mode collapses and seeds a pair Alfvén waves. An intermediate B₀ couples the thermal anisotropy to the pair Alfvén mode and propagating magnetowaves grow. The phase speed of the pair of Alfvén waves is increased by the thermal anisotropy. Its growth is suppressed when B₀ is set to the value that stabilizes the mirror mode.