  This article deals with the study of electromagnetic waves equations and the
Lorentz condition, as emergent properties of Maxwell's system in the context of
systems theory. To do this, the wave equations and the Helmholtz equation are
first deduced. Using the displaced Dirac operator, which is closely related to
the main vector calculation operators, it is possible to establish a direct
connection between the solutions of the Maxwell time-harmonic system and two
quaternion equations. Also, the application of the Lorentz condition to
transform the time-harmonic Maxwell system into a simple quaternion equation
based on the scalar and vector potentials is exposed.
