Non-intersecting splitting algebras in a non-Bernoulli transformation

Given a measure preserving transformation $T$ on a Lebesgue $σ$ algebra, a complete $T$ invariant sub $σ$ algebra is said to split if there is another complete $T$ invariant sub $σ$ algebra on which $T$ is Bernoulli which is completely independent of the given sub $σ$ algebra and such that the two sub $σ$ algebras together generate the entire $σ$ algebra. It is easily shown that two splitting sub $σ$ algebras with nothing in common imply $T$ to be K. Here it is shown that $T$ does not have to be Bernoulli by exhibiting two such non-intersecting $σ$ algebras for the $T,T^{-1}$ transformation, negatively answering a question posed by Thouvenot in 1975.

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