Summary: Verstraete, Dehaene and DeMoor showed that each of the two-qubit states can be generated from one of two canonical families of two-qubit states by means of transformations preserving the tensor structure of the state space. Precisely, each of such states can be generated from a three-parameter family of Bell-diagonal states or from three-parameter rank-deficient states. In this paper, we show that this classification of two-qubit states can be refined. In particular, we show that the latter canonical family of states can be reduced to three fixed states and a two-parameter family of two-qubit states. For this family of states, we provide a simple parametrization that guarantees positive semidefiniteness of the states and enables easier calculation of the Wootters concurrence and quantum discord. Moreover, we present a new general parametrization of all two-qubit states generated from the canonical families of states using sets of (pseudo)orthogonal four-vectors (frames). An advantage of the presented approach lies in the fact that the standard conditions for positive semidefiniteness of states are equivalent to (pseudo)orthogonality conditions for four-vectors serving as parameters (and appropriate conditions for parameters of the corresponding canonical family of states).