Understanding the nature of the mysterious pseudogap phenomenon is one of the most important issues associated with cuprate high-$T_c$ superconductors. Here, we report $^{17}$O nuclear magnetic resonance (NMR) studies on two planar oxygen sites in stoichiometric cuprate YBa$_2$Cu$_4$O$_8$ to investigate the symmetry breaking inside the pseudogap phase. We observe that the Knight shifts of the two oxygen sites are identical at high temperatures but different below $T_{rm nem} sim$ 185 K, which is close to the pseudogap temperature $T^{ast}$. Our result provides a microscopic evidence for intra-unit-cell electronic nematicity. The difference in quadrupole resonance frequency between the two oxygen sites is unchanged below $T_{rm nem}$, which suggests that the observed nematicity does not directly stem from the local charge density modulation. Furthermore, a short-range charge density wave (CDW) order is observed below $T simeq$ 150 K. The additional broadening in the $^{17}$O-NMR spectra because of this CDW order is determined to be inequivalent for the two oxygen sites, which is similar to that observed in case of nematicity. These results suggest a possible connection between nematicity, CDW order, and pseudogap.
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