Multiqubit positive-partial-transpose (PPT) entangled states provide canonical examples of bound entanglement and stringent test cases for multipartite separability criteria. We study PPT entangled states of the minimum possible rank in the three-qubit system, namely rank four. In the established terminology, states with a nonzero Lorentz invariant are called type I, whereas those with a zero Lorentz invariant are called type II. Every such state constructed from an unextendible product basis (UPB) has a nonzero Lorentz invariant and hence is of type I. We give a necessary and sufficient criterion for determining whether a type-I state is SLOCC-equivalent to a PPT entangled state constructed from a UPB. For type-II states, we obtain a Bell-basis normal form and prove that it exhausts all such states up to SLOCC equivalence. We further give a necessary and sufficient condition for two states in this normal form to be SLOCC-equivalent. We also determine the range of the Lorentz invariant for multiqubit states.