We have the Dirac equation for spin-1/2 particles [1-5]
is a 4-spinor with the
matrices verifying the anticommutator [6-8]:
Here we shall use the Dirac-Pauli (or standard) representation [2,9]:
with the Cayley -Sylvester -Pauli  matrices:
to analyze the transformation law of under the orthochronic and proper Lorentz group [13-19]:
which implies the existence [2, 7, 20, 21] of a non-singular matrix S such that:
and we deduce the relativistic invariance of (1) if the Dirac 4-spinor obeys the transformation rule:
Here we exhibit a method to findS for a given Lorentz matrix.
Construction of the matrix S for a given Lorentz mapping.
The arbitrary complex quantities α, β, γ, δ verifying the constraint αδ – βγ = 1, generate a Lorentz matrix
via the relations [13-15, 17, 22-26]:
where cc means the complex conjugate of all the previous terms.
The inverse problem is to obtain α, β, γ, δ if we know L, and the answer is [26-29]:
From (6) are immediate the expressions [3, 30]:
that is, if we know S then with (10) we can determine the Lorentz matrix; (10) generates the relations:
which allow to obtain L if we have the expansion :
However, here we have the inverse problem, that is, to obtain
verifying (11) for a given Lorentz matrix. Our answer is the following:
hence the expressions (8) are deduced if we apply (13) into (11). Besides, with (13) the matrix (12) acquires the structure:
Therefore, for a given Lorentz transformation first we employ (9) to determine α,β,γ,δ, then S is immediate via (14); this approach is an alternative to the process showed in  and to the explicit general formula obtained by Macfarlane :
however, the possible physical applications are not evident in it. In our procedure, for example, the relations (9) are of great interest for the physicists working on supersymmetry , and the expressions (14) are very useful to study the relativistic motion of a classical point particle .
The Dirac equation is relativistic if the corresponding 4-spinor verifies the transformation (7) under Lorentz mappings, with the matrix S satisfying the condition (6). Here we showed a procedure to construct S for a given Lorentz matrix.