Integrable distributions are involutive

By integrability, the point $p$ lies on a connected integral manifold [given] $i:N\to M$ of $\mathcal{D}$ having the same dimension as the distribution. After shrinking near the point of $N$ over $p$, the immersion may be viewed as an embedding, so $X$ and $Y$ restrict to smooth vector fields $\tilde{X}$ and $\tilde{Y}$ on that local piece of $N$.

Along that local integral manifold, the fields $\tilde{X}$ and [given] $\tilde{Y}$ are $i$-related to $X$ and $Y$. Therefore their Lie bracket is $i$-related to $[X,Y]$. Since the bracket on $N$ is tangent to $N$, the value $[X,Y{]}_p$ lies in the image of $di$, which is ${\mathcal{D}}_p$.

The point $p$ and the tangent fields $X,Y$ were arbitrary, so [given] $[X,Y]\in \mathrm{\Gamma }(\mathcal{D})$. Hence $\mathcal{D}$ is involutive.