  It is shown that a simple continuity condition in the algebra of split
octonions suffices to formulate a system of differential equations that are
equivalent to the standard Dirac equations. In our approach the particle mass
and electro-magnetic potentials are part of an octonionic gradient function
together with the space-time derivatives. As distinct from previous attempts to
translate the Dirac equations into different number systems here the wave
functions are real split octonions and not bi-spinors. To formulate positively
defined probability amplitudes four different split octonions (transforming
into each other by discrete transformations) are necessary, rather then two
complex wave functions which correspond to particles and antiparticles in usual
Dirac theory.
