  We examine the SLOCC classification of the (non-normalized) pure states of
four qubits obtained by F. Verstraete et al. The rigorous proofs of their basic
results are provided and necessary corrections implemented. We use Invariant
Theory to solve the problem of equivalence of pure states under SLOCC
transformations of determinant 1 and qubit permutations. As a byproduct, we
produce a new set of generators for the invariants of the Weyl group of type
F_4. We complete the determination of the tensor ranks of 4-qubit pure states
initiated by J.-L. Brylinski. As a result we obtain a simple algorithm for
computing these ranks. We obtain also a very simple classification of pure
states of rank at most 3.
