Motivated by the  [following  answer](https://mathoverflow.net/questions/193650/a-vector-space-associated-with-a-vector-field-on-a-symplectic-manifold/303162#303162)  we  consider the  following  Lie  agebra  associated to  a  foliation:

Let  $\mathcal{F}$  be  a  foliation of  a  manifold  $M$  tangent to  integrable subbundle $D$  of  $TM$.  We define  the  following  Lie  algebra  of  vector  fields on $M$:

$$A_{\mathcal{F}}=\{Y\in \chi^{\infty}(M)\mid [X,Y]\;\; \text{is  tangent to $\mathcal{F}$} \text{ for  all } X\in \Gamma (D)\} $$

In  fact $A_{\mathcal{F}}$ is  the  idealizer of  the  Lie  algebra $L_{\mathcal{F}}$  of  vector  fields on $M$ which are tangent to the  foliation.

>First  Question: Is there a  foliaion for  which  $A_{\mathcal{F}}=L_{\mathcal{F}}$?  

>Is it true  to say that the  dimension of  $\{Y(p)\mid Y \in A_{\mathcal{F}}\}\subseteq T_pM$ is  independent of  Choosing $p\in M$. 


If the  answer  of the  second  question is yes, then $A_{\mathcal{F}}$  defines an integrable   distribution $D'$ containing the previous one, i.e. $D$. It  generates  a  foliation $\mathcal{F}'$ which would be  defined  as  saturation of  $\mathcal{F}$. 


>Are there some  example of  this  situation such that $dim(\mathcal{F}) <dim({\mathcal{F}')<dim(M)}$?