  Upon straightforward four--directional extension of the special--relativistic
two--dimensional transformations to the four--dimensional case we lead to
convenient totally anisotropic kinematic transformations, which prove to reveal
many remarkable group and invariance properties. Such a promise is shown to
ground the basic manifold with the Finslerian fourth-root metric function to
measure length of relativistic four--vectors. Conversion to the framework of
relativistic four--momentum is also elucidated. The relativity principle is
strictly retained. An interesting particular algebra for subtraction and
composition of three-dimensional relative velocities is arisen. The
correspondence principle is operative in the sense that at small relative
velocities the transformations introduced tend approximately to ordinary
Lorentzian precursors. The transport synchronization remains valid.
  Abbreviation RF will be used for (inertial) reference frames.
  {\bf Keywords:} special relativity, invariance, Finsler geometry.
