The estimation of the diffusion matrix $Σ$ of a high-dimensional, possibly time-changed Lévy process is studied, based on discrete observations of the process with a fixed distance. A low-rank condition is imposed on $Σ$. Applying a spectral approach, we construct a weighted least-squares estimator with nuclear-norm-penalisation. We prove oracle inequalities and derive convergence rates for the diffusion matrix estimator. The convergence rates show a surprising dependency on the rank of $Σ$ and are optimal in the minimax sense for fixed dimensions. Theoretical results are illustrated by a simulation study.