  Using the variational characterization of the principal (i.e., smallest)
eigenvalue below the essential spectrum of a lower semibounded self-adjoint
operator, we prove strict domain monotonicity (with respect to changing the
finite interval length) of the principal eigenvalue of the Friedrichs extension
$T_F$ of the minimal operator for regular four-coefficient Sturm--Liouville
differential expressions. In the more general singular context, these
four-coefficient differential expressions act according to \[
  \tau f = \frac{1}{r} \left( - \big(f^{[1]}\big)' + s f^{[1]} +
qf\right)\,\text{ with $f^{[1]} = p [f' + s f]$ on $(a,b) \subseteq
\mathbb{R}$}, \] where the coefficients $p$, $q$, $r$, $s$ are real-valued and
Lebesgue measurable on $(a,b)$, with $p > 0$, $r>0$ a.e.\ on $(a,b)$, and
$p^{-1}$, $q$, $r$, $s \in L^1_{loc}((a,b); dx)$, and $f$ is supposed to
satisfy \[ f \in AC_{loc}((a,b)), \; p[f' + s f] \in AC_{loc}((a,b)). \] This
setup is sufficiently general so that $\tau$ permits certain distributional
potential coefficients $q$, including potentials in $H^{-1}_{loc}((a,b))$.
  As a consequence of the strict domain monotonicity of the principal
eigenvalue of the Friedrichs extension in the regular case, and on the basis of
oscillation theory in the singular context, in our main result, we characterize
all lower bounds of $T_F$ as those $\lambda\in \mathbb{R}$ for which the
differential equation $\tau u = \lambda u$ has a strictly positive solution $u
> 0$ on $(a,b)$.
