  We present a product of pairs of pointed Hilbert spaces that, in the context
of Boz\.ejko, Leinert and Speicher's theory of conditionally free probability,
plays the role of the reduced free product of pointed Hilbert spaces, and thus
gives a unified construction for the natural notions of independence defined by
Muraki.
  We additionally provide important applications of this construction. We prove
that, assuming minor restrictions, for any pair of conditionally free algebras
there are copies of them that are conditionally free and also free, a property
that is frequently assumed (as hypothesis) to prove several results in the
literature. Finally, we give a short proof of the linearization property of the
$^cR$-transform (the analog of Voiculescu's $R$-transform in the context of
conditionally free probability).
