A lemma on Lie bracket under insufficient smoothness

Let two vector fields on a $C^2$-variety $M$ be tangent to a $C^1$-submanifold $F\subset M$. We show if that these fields are differentiable at a point $p\in F$, then their Lie bracket is also tangent to $F$. This statement is a weakening of the “easy part” assumptions of the Frobenius theorem.