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Using a noncommutative Lyapunov theorem of Akemann and Weaver, we show that finite dimensional orthogonality constraints can be realized by projections, and hence by symmetries. Iterating this construction, we prove that if $M$ is a diffuse finite von Neumann algebra with faithful normal tracial state $\\tau$ and $L^2(M,\\tau)$ is separable, then $L^2(M,\\tau)$ admits an orthonormal basis consisting of self-adjoint unitaries in $M$. 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