Non-intersecting splitting σ-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.