Recent studies on the phase-space dynamics of a one-dimensional Lennard-Jones fluid reveal the existence of regular collective perturbations associated with the smallest positive Lyapunov exponents of the system, called hydrodynamic Lyapunov modes, which previously could only be identified in hard-core fluids. In this work we present a systematic study of the Lyapunov exponents and Lyapunov vectors, i.e. perturbations along each direction of phase space, of a three-dimensional Lennard-Jones fluid. By performing the Fourier transform of the spatial density of the coordinate part of the Lyapunov vector components and then time-averaging this result we find convincing signatures of longitudinal modes, with inconclusive evidence of transverse modes for all studied densities. Furthermore, the longitudinal modes can be more clearly identified for the higher density values. Thus, according to our results, the mixing of modes induced by both the dynamics and the dimensionality induces a hitherto unknown type of order in the tangent space of the model herein studied at high density values.