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 J$J$ such that J²=0$J^2=0$ and for every pair of vector fieldsX$X$,Y$Y$ on M: [JX,JY]−J[JX,Y]−J[X,JY]+J²[X,Y]=0$[JX,JY]-J[JX,Y]-J[X,JY]+J^2[X,Y]=0$. For every open set Ω$\Omega$ of V, J. Lehmann-Lejeune studied the Lie Algebra _LJ(Ω)$L_J\left(\Omega \right)$ of vector fields X defined on Ω$\Omega$ such that the Lie derivative L(X)J$L\left(X\right)J$ is equal to zero i.e., for each vector field Y$Y$on Ω$\Omega$: [X,JY]=J[X,Y]$[X,JY]=J[X,Y]$ and showed that for every vector field X on Ω$\Omega$ such thatX∈KerJ$X\in KerJ$, we can write X=∑[Y,Z]$X=\sum [Y,Z]$ where ∑$\sum$is a finite sum and Y,Z$Y,Z$ belongs to _LJ(Ω)∩(Ker_J|Ω)$L_J\left(\Omega \right)\cap \left(Ker{J}_{|\Omega }\right)$.