Differential equations for the Majorana particle in $3+1$ and $1+1$ dimensions

The relativistic wave equation considered to mathematically describe the Majorana particle is the Dirac equation with a real Lorentz scalar potential plus the 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 nonequivalent 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 standpoint, 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. In addition, we rederive and use results that follow from a procedure already introduced by Case to obtain a two-component Majorana equation in $3+1$ dimensions. We for the first time introduce a similar procedure in $1+1$ dimensions and then use the obtained results.