  In these notes we first review Pauli's proof of his `fundamental theorem'
that states the equivalence of any two sets of Dirac matrices $\{ \gamma^\mu
\}$. Due to this theorem not only all physical results in the context of the
Dirac equation have to be independent of the basis chosen for the Dirac
matrices, but it should also be possible to obtain the results without
resorting to a specific basis in the course of the computation. Indeed, we
demonstrate this in the case of the behaviour of Dirac spinors under Lorentz
transformations, the quantization of the Dirac field, the expectation value of
the spin operator and several other topics. In particular, we emphasize the
totally different physics and mathematics background of the matrix $\beta$,
used in the definition of the conjugate Dirac spinor, and $\gamma^0$. Finally,
we compare the basis-independent manipulations with those performed in the Weyl
basis of Dirac matrices. The present notes provide a self-contained
introduction to the Dirac theory by solely exploiting the simplicity of the
Dirac algebra and the power of Pauli's Theorem.
