Lie algebra on the transverse bundle of a decreasing family of foliations

J. Lehmann-Lejeune in [J. Lehmann-Lejeune, Cohomologies sur le fibré transverse à un feuilletage, C.R.A.S. Paris 295 (1982), 495–498] defined on the transverse bundle V to a foliation on a manifold M, a zero-deformable structure JJ such that J2=0J2=0 and for every pair of vector fieldsXX,YY on M: [JX,JY]−J[JX,Y]−J[X,JY]+J2[X,Y]=0[JX,JY]−J[JX,Y]−J[X,JY]+J2[X,Y]=0. For every open set ΩΩ of V, J. Lehmann-Lejeune studied the Lie Algebra LJ(Ω)LJ(Ω) of vector fields X defined on ΩΩ such that the Lie derivative L(X)JL(X)J is equal to zero i.e., for each vector field YYon ΩΩ: [X,JY]=J[X,Y][X,JY]=J[X,Y] and showed that for every vector field X on ΩΩ such thatX∈KerJX∈KerJ, we can write X=∑[Y,Z]X=∑[Y,Z] where ∑∑is a finite sum and Y,ZY,Z belongs to LJ(Ω)∩(KerJ|Ω)LJ(Ω)∩(KerJ|Ω).In this note, we study a generalization for a decreasing family of foliations.