Atomic mechanism of glass formation of a supercooled simple monatomic liquid with Lennard-Jones-Gauss (LJG) interatomic potential is studied by molecular dynamics (MD) simulation. Supercooled and glassy states are obtained by cooling from the melt. Glassy state obtained at low temperatures is annealed for very long time, on the order of microsecond, and we find that glassy state remains unchanged and that the long-lived glassy state of a simple monatomic system in three dimensions is realized. We analyze the spatiotemporal properties of solid-like and liquid-like atoms that are defined by the Lindemann-like freezing criterion. The number of solid-like atoms, distributed throughout the liquid, increases with decreasing temperature toward glass transition and they form clusters. In the deeply supercooled region, almost all solid-like atoms form a single percolation cluster and its characteristic size increases sharply on further cooling. Glass formation in supercooled liquid occurs when a single percolation cluster of solid-like atoms involves a majority of atoms to form a relatively rigid solid phase. We also obtain several physical quantities of the system, including temperature dependence of mass density, Lindemann ratio, incoherent intermediate scattering function, α-relaxation time, evolution of radial distribution function, and local bond-pair orders detected by Honeycutt-Andersen analysis. We identify three characteristic temperatures related to the vitrification: a temperature at which crossover from liquid-like to solid-like dynamics occurs on cooling, the glass transition temperature, and the Vogel-Fulcher-Tammann temperature. Behavior of liquid-like atoms in glassy state has been analyzed and discussed.