The standard momentum operator −i∇ has the trivial domain (the null vector) if the L2 Hilbert space consists of only real-valued functions. In consequence, it is useless in quantum mechanics of the relativistic Majorana particle which is formulated in such a Hilbert space. Instead, one can consider the axial momentum operator introduced in (2019) Phys. Lett. A 383 1242. In the present paper we report several new results which elucidate usability of the axial momentum observable. First, a new motivation for the axial momentum is given, and the Heisenberg uncertainty relation checked. Next, we show that the general solution of time evolution equation written in the axial momentum basis has a connection with quaternions. Furthermore, it turns out that in the case of massive Majorana particles, single traveling monochromatic plane waves are not possible, but there exist solutions which have the form of two plane waves traveling in opposite directions. Another issue discussed here in detail is relativistic invariance. A single real, orthogonal and irreducible representation of the Poincaré group—consistent with the lack of antiparticle—is unveiled.