Let $M$ be a smooth $d$-dimensional Riemannian manifold, and let $p \in M$.
Does there always exist an open neighbourhood $U$ of $p$, and an orthonormal frame $E_i$ over $U$, s.t $\text{span}\{ E_1,\dots,\hat E_i,\dots E_d\} $ is involutive for every $i$?
Does it help if we only require $\text{span}\{ E_1,\dots,\hat E_i,\dots E_d\} $ to be involutive for a single $i$?