  In general, the relativistic wave equation considered to mathematically
describe the so-called Majorana particle is the Dirac equation with a real
Lorentz scalar potential plus the so-called Majorana condition. Certainly,
depending on the representation that one uses, the resulting differential
equation changes. It could be a real or a complex system of coupled equations,
or it could even be a single complex equation for a single component of the
entire wave function. Any of these equations or systems of equations could be
referred to as a Majorana equation or Majorana system of equations because it
can be used to describe the Majorana particle. For example, in the Weyl
representation, in (3+1) dimensions, we can have two non-equivalent explicitly
covariant complex first-order equations; in contrast, in (1+1) dimensions, we
have a complex system of coupled equations. In any case, whichever equation or
system of equations is used, the wave function that describes the Majorana
particle in (3+1) or (1+1) dimensions is determined by four or two real
quantities. The aim of this paper is to study and discuss all these issues from
an algebraic point of view, highlighting the similarities and differences that
arise between these equations in the cases of (3+1) and (1+1) dimensions in the
Dirac, Weyl, and Majorana representations. Additionally, to reinforce this
task, we rederive and use results that come from a procedure already introduced
by Case to obtain a two-component Majorana equation in (3+1) dimensions.
Likewise, we introduce for the first time a somewhat analogous procedure in
(1+1) dimensions and then use the results we obtain.
