Euclidean rotations in $\mathbb{R}^n$ are bijective and isometric maps. Nevertheless, they lose these properties when digitized in $\mathbb{Z}^n$. For $n=2$, the subset of bijective digitized rotations has been described explicitly by Nouvel and R\'emila and more recently by Roussillon and C{\oe}urjolly. In the case of 3D digitized rotations, the same characterization has remained an open problem. In this article, we propose an algorithm for certifying the bijectivity of 3D digitized rational rotations using the arithmetic properties of the Lipschitz quaternions.