Tangency under the Lie bracket
= Tangency under the Lie bracket
If $X$ and $Y$ preserve $I(N)$, then their commutator preserves it because
$$
[X,Y]f=X(Yf)-Y(Xf)\in I(N)
$$
for every $f\in I(N)$. Thus vector fields tangent to an embedded submanifold are closed under the <Lie bracket of vector fields>, and the restricted bracket depends only on their restrictions to the submanifold.