Non-intersecting splitting algebras in a non-Bernoulli transformation

Given a measure preserving transformation $T$ on a Lebesgue $\sigma$ algebra, a complete $T$ invariant sub $\sigma$ algebra is said to split if there is another complete $T$ invariant sub $\sigma$ algebra on which $T$ is Bernoulli which is completely independent of the given sub $\sigma$ algebra and such that the two sub $\sigma$ algebras together generate the entire $\sigma$ algebra. It is easily shown that two splitting sub $\sigma$ 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 $\sigma$ algebras for the $T,T^{-1}$ transformation, negatively answering a question posed by Thouvenot in 1975.