Motivated by the following answer 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 initial distribution $D$. It generates a foliation $\mathcal{F}'$ which would be defined as saturation of $\mathcal{F}$.

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

Remark: As we see in the linked question, when we have a foliation $\mathcal{F}$ of a surface $M$ which is tangent to a nonvanishing vector field $X$, then the Lie algebra $A_{\mathcal{F}}$ is equal to $\{Y\mid [X,Y]=fX,\;\text{for some }f\in C^{\infty}(M)\}$.