  The gauge field strength tensor $F_{\mu \nu}$ in Abelian and non-Abelian
local gauge field theories is a key object in the construction of the
Lagrangian since it provides the kinetic term(s) of the gauge field(s) $A_\mu$.
When introducing this object, most of textbooks employ as a tool the commutator
of the gauge covariant derivatives $D_\mu \psi$ of a fermion field $\psi$:
$F_{\mu \nu} \psi = (i/g)[D_\mu,D_\nu]\psi$. I argue that such a construction,
although completely correct and valid, is not pedagogical enough for many
students that approach the gauge field theories for the first time. Another
construction, based on the object $D_\mu A_\nu$: $F_{\mu \nu} = D_{[\mu}
A_{\nu]}$, which avoids the introduction of additional and, for the case in
consideration, spurious degrees of freedom such as the fermion one, simpler,
more pedagogical in many cases, and suitable for first-time students, is
presented.
