Inspired by the Lie algebra discussed [in this 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 given 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 foliation $\mathcal{F}$ 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 $\{Y\mid [X,Y]=fX,\;\text{for some }f\in C^{\infty}(M)\}$.