The quest for identification and understanding of fractional vorticity is a major subject of research in the quantum fluids domain, ranging from superconductors, superfluid Helium-3 to cold atoms. In a two-dimensional Bose degenerate gas with a spin degree of freedom, the fundamental topological excitations are fractional vortical entities, called half-quantum vortices. Convincing evidence for the existence of half-quantum vortices was recently provided in spinor polariton condensates. The half-quantum vortices can be regarded as the fundamental structural components of singly charged vortices but, so far, no experimental evidence of this relation has been provided. Here we report on the direct and time-resolved observation of the dynamical process of the dissociation of a singly charged vortex into its primary components, a pair of half-quantum vortices. The physical origin of the observed phenomenology is found in a spatially inhomogeneous static potential that couples the two spin components of the condensate.