Let $f,g$ two smooth vector fields on $\mathbb{R}^n$ tangent to a surface $S$ at every point. I think it implies that the Lie bracket $[f,g]$ is tangent to $S$.
I start from the fact that $$ [f,g](x) = \lim_{e\to 0} \frac{1}{e^2}\left[\exp(ef)\exp(eg) - \exp(eg)\exp(ef) \right]x $$
Since, for $s$ small enough, the curve $s\to (\exp(sf)\exp(sg) - \exp(sg)\exp(sf))(x)$ is contained in $S$ this proves that $[f,g]$ is tangent to $S$.
am I missing something ?