On two maximally entangled couples

In a seminal article, Higuchi and Sudbery showed that a pure four-qubit state can not be maximally entangled across every bipartition. Such states are now known as absolutely maximally entangled (AME) states. Here we give a series of old and new proofs of the fact that no four-qubit AME state exists. These are based on invariant theory, methods from coding theory, and basic properties from linear algebra such as the Pauli commutation relations.