  In the time-reversal-breaking centrosymmetric systems, the appearance of Weyl
points can be guaranteed by an odd number of all the even/odd parity occupied
bands at eight inversion-symmetry-invariant momenta. Here, based on symmetry
analysis and first-principles calculations, we demonstrate that for the
time-reversal-invariant systems with $S_4$ symmetry, the Weyl semimetal phase
can be characterized by the inequality between a well-defined invariant $\eta$
and an $S_4$ indicator $z_2$. By applying this criterion, we find that some
candidates, previously predicted to be topological insulators, are actually
Weyl semimetals in the noncentrosymmetric space group with $S_4$ symmetry. Our
first-principles calculations show that four pairs of Weyl points are located
in the $k_{x,y}$ = 0 planes, with each plane containing four same-chirality
Weyl points. An effective model has been built and captures the nontrivial
topology in these materials. Our strategy to find the Weyl points by using
symmetry indicators and invariants opens a new route to search for Weyl
semimetals in the time-reversal-invariant systems.
