Lemma 2 ([Popa's ultrapower freeness]). Let P be a II₁ factor and let B⊂P^ω be a unital von Neumann subalgebra with separable predual. There is a diffuse abelian von Neumann subalgebra of P^ω free from B . Consequently, for every d∈ , …

Apply Theorem~0.1(a) of with every coordinate algebra and every coordinate subalgebra equal to P . The required non-intertwining condition holds: a nonzero corner of the diffuse factor P cannot embed in the scalar relative commutant, or in a finite matrix amplification of it. Moreover, P^ω is a factor, so the relative commutant of the full ultraproduct subalgebra is 1 . The conclusion for the separable centered subspace B⊖1 is therefore ordinary scalar-trace freeness. This is the case in which every coordinate subalgebra equals its ambient factor, discussed after that theorem and in . A diffuse abelian tracial algebra contains a Haar unitary. Choose one, adjoin it to B , and repeat. Finite adjunction preserves separable predual: a countable generating set gives a separable tracial Hilbert space by polynomial approximation. Thus all d choices take place in the same ultrapower. At each step the new algebra is free from the algebra generated at the preceding steps, giving the asserted family.