Is a subspace containing a splitting subspace a splitting subspace?

If V, A and B are three closed subspaces of L2. ; F ; P/ we say that V is a splitting subspace for A; B if and only if A and B are conditionally orthogonal given V. If V is a splitting subspace for A; B, we shall say that V splits A; B. Rozanov [Rozanov, Y.A., 1979. Stochastic Markovian Fields. In: Developments in Statistics, vol. 2. Academic Press, New York, p. 205] observes that A ? BjV does not imply that the closed subspace W V splits A; B. However, no example is provided. In this note we provide one.