Based on irreducible representations (or symmetry eigenvalues) and compatibility relations (CR),a material can be predicted to be a topological/trivial insulator (satisfying CR) or a topological semimetal (violating CR).However,Weyl semimetals (WSMs) usually go beyond this symmetry-based strategy.In other words,Weyl nodes could emerge in a material,no matter if its occupied bands satisfy CR,or if the symmetry indicators are zero.In this work,we propose a new topological invariant x for the systems with S4 symmetry (i.e.,the improper rotation S4(≡ IC4z) is a proper fourfold rotation (C4z) followed by inversion (Ⅰ)),which can be used to diagnose the WSM phase.Moreover,x can be easily computed through the onedimensional Wilson-loop technique.By applying this method to the high-throughput screening in our first-principles calculations,we predict a lot of WSMs in both nonmagnetic and magnetic compounds.Various interesting properties (e.g.,magnetic frustration effects,superconductivity and spin-glass order,etc.) are found in predicted WSMs,which provide realistic platforms for future experimental study of the interplay between Weyl fermions and other exotic states.