Frobenius local coordinate theorem

If $\mathcal{D}$ is integrable, then it is involutive by the necessity [given] proposition. This proves 1 => 2.

Now assume $\mathcal{D}$ is involutive. For $k=0$ the distribution is [given] zero, and for $k=n$ it is all of $TM$, so the displayed coordinate form is immediate. Thus only the case $1\le k<n$ needs work.

Choose a local frame $X_1,\dots ,X_k$ of $\mathcal{D}$ near $p$ with [given] $X_1(p)\mathrm{\ne }0$. By the frame-reduction lemma, after shrinking there are local sections $Y_2,\dots ,Y_k$ such that $X_1,Y_2,\dots ,Y_k$ frames $\mathcal{D}$, each $Y_j$ is tangent to the slices of a flow-box chart for $X_1$, and $[X_1,Y_j]=0$ for all $j\ge 2$. Let $S$ be the slice $x^1=0$ in that flow-box chart, and write $V_j:=Y_j{\mathrm{\mid }}_S$. Then $V_2,\dots ,V_k$ are pointwise independent vector fields on the $(n-1)$-manifold $S$. Because $[Y_i,Y_j]\in \mathrm{\Gamma }(\mathcal{D})$ and each $Y_j$ is tangent to the slices, the restrictions $[V_i,V_j]=[Y_i,Y_j]{\mathrm{\mid }}_S$ lie in the span of $V_2,\dots ,V_k$. Hence those $V_j$ span an involutive rank-$(k-1)$ distribution on $S$.

Apply the theorem inductively on the rank to that distribution on $S$. [given] There are local coordinates $(x^2,\dots ,x^k,x^{k+1},\dots ,x^n)$ on $S$ in which $\mathrm{span}(V_2,\dots ,V_k)=\mathrm{span}({\mathrm{\partial }}_{x^2},\dots ,{\mathrm{\partial }}_{x^k})\mathrm{.}$ Extend these coordinates off $S$ by keeping them constant along the $X_1$-flow, and use the flow parameter as $x^1$. Then $X_1={\mathrm{\partial }}_{x^1}$. Since each $Y_j$ commutes with $X_1$, its coefficients in these flow-box coordinates are constant along the $X_1$-flow, so the span identity on $S$ extends to $\mathrm{span}(Y_2,\dots ,Y_k)=\mathrm{span}({\mathrm{\partial }}_{x^2},\dots ,{\mathrm{\partial }}_{x^k})\mathrm{.}$ Therefore $\mathcal{D}=\mathrm{span}({\mathrm{\partial }}_{x^1},\dots ,{\mathrm{\partial }}_{x^k})$ on a neighborhood of $p$.

In those coordinates, the slices with [given] $x^{k+1},\dots ,x^n$ fixed are integral manifolds of $\mathcal{D}$. Thus the involutive case is integrable, proving 2 => 1.

Hence integrability and involutivity are equivalent, and in the involutive [given] case the distribution is locally flat in coordinates as stated.