Common formulations of the eddy current problem involve either vector or scalar potentials, each with its own advantages and disadvantages. An impasse arises when using scalar potential-based formulations in the presence of conductors with non-trivial topology. A remedy is to augment the approximation spaces with generators of the first cohomology group. Most existing algorithms for this require a special, e.g., hierarchical, finite element basis construction. Using insights from de Rham complex approximation with splines, we show that additional conditions are here unnecessary. Spanning tree techniques can be adapted to operate on a hexahedral mesh resulting from isomorphisms between spline spaces of differential forms and de Rham complexes on an auxiliary control mesh.