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A constant-rank smooth distribution $D$ is an integrable distribution exactly when it is involutive. In differential-form language, let ${\alpha }^1,\dots ,{\alpha }^q$ be a local basis of its annihilator, so $D={\bigcap }_i\ker {\alpha }^i$. The equivalent condition isfor suitable local one-forms ${\beta }^i{}_j$. More precisely, the ideal generated by the annihilating forms inside the exterior algebra of differential forms is closed under the exterior derivative; this is an exterior differential ideal. It is not a claim that a two-form lies in the one-form annihilator itself.

For local vector fields $X,Y$ tangent to $D$, the definition of exterior derivative givesThus the displayed closure condition says exactly that the Lie bracket of vector fields remains tangent to $D$. The theorem turns this local algebraic condition into integral manifolds through every point.

For a nowhere-zero one-form $\alpha$, its kernel is tangent to local hypersurfaces precisely when $\alpha \wedge d\alpha =0$. In general relativity, applying this to a metric-dual one-form characterizes hypersurface orthogonality and vanishing twist.