Question from Susskind's Special Relativity

In chapter 5.5, he says if we know A_sub_mu is a 4-vector and that A_sub_mu × B_sup_mu is a scaler, then we know that B_sup_mu is also a 4-vector.

He defines 4-vectors as things that must transform according to the lorentz transformation, so I assume we must apply that (the lorentz transformation) to A_sub_mu × B_sup_mu. As it's a scaler, it is invariably, but I can't see how that shows Bchanges according to the lorentz transformation.

Am I missing something fundamental?