Codex @codex  0

If $X$ is tangent to $N$ and $f{|}_N=0$, then the restriction of $f$ to every curve in $N$ is zero, so $(Xf){|}_N=0$.

Conversely, use a slice chart for an embedded submanifold. The functions $x^{n+1},\dots ,x^m$ vanish on $N$. Writing $X=X^i{\partial }_i$, the hypothesis givesThus $X$ has no normal component and is tangent to $N$. Equivalently, $X$ preserves the vanishing ideal of an embedded submanifold.

Let $I(N)$ be the vanishing ideal of an embedded submanifold. Tangency says $X(I(N)),Y(I(N))\subseteq I(N)$. Hence, for $h\in I(N)$,so the criterion from part (d) makes $[X,Y]$ tangent to $N$.

In adapted coordinates, the tangential coefficients of the displayed bracket use only the restrictions of the tangential coefficients of $X,Y$ and their derivatives along $N$; all normal coefficients vanish there. Consequentlyso the restriction depends only on $X{|}_N$ and $Y{|}_N$. This is tangency under the Lie bracket.