Feuilletages transversalement tangents

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A foliation ℱ on a differentiable manifold M is said to be transversally tangent if the jacobian matrices of the change of the transverse coordinates are in the tangent group. Such foliation exists if and only if there is an endomorphism J of its normal bundle such that J2=0 and such that the Nijenhuis tensor of J is the zero-tensor. In the special case of the tangent bundle of order 2, T2M, its total space has a natural (1, 1)-tensor F such that F3=0 and an integrable almost-tangent structure. We study several Lie algebras associated to these structures.