What is the range of the Levi-Civita connection? Let $E$  a distribution in $TQ$,   $Q$ a riemannian manifold.  Let $X,Y$  vectorfields
tangent to  $E$.
Question:  is it true that   $\nabla_X Y  \in  E + [E,E]$.
If not give a counterexample.
I would be content to know true or false for  X=Y.   This is motivated by
a short paper by Cartan n nonholonomic mechanics  at the ICM 1928. The arguments
he uses seem to imply this conclusion, although in a roundabout way. Cartan shows that one can change the metric at will outside E+[E,E] without changing the NH geodesics.
By looking at Cartan's arguments,  it seems that the result holds for any torsion free connection in TQ, no need to be compatible with a metric.  
One chooses a  frame with 3 types of vectorfields,
i) first ones tangent to E, ii) second group forming a complement of E in
E + [E,E] , iii) complete  to TQ .  One considers the dual coframe and writes the structure equations.  The third group in the coframe is the derived ideal of
the ideal of forms that annihilate E.  
I believe that writing the covariant derivative nabla_X Y  for  X,Y in E  using the structure forms, one gets the desired result.
 A: Let $Q = SU(2)$, and consider the diagonal maximal torus $U(1)$ as acting freely and isometrically on $SU(2)$ by multiplication from the right, which therefore gives rise to the Hopf fibration $SU(2) \to SU(2)/U(1) = \mathbb{C}P^1$. Let $E = VSU(2)$ be the vertical tangent bundle of this submersion, which is integrable, so that $E + [E,E] = E$. Finally, let $g_0$ be the $\operatorname{Ad}$-invariant Riemannian metric on $SU(2)$ induced by the Killing form, let $\ell \in C^\infty(SU(2),(0,+\infty))^{U(1)}$ be non-constant, and let $g = \ell g_0$ be the resulting $U(1)$-invariant Riemannian metric on $Q = SU(2)$, so that $SU(2) \to \mathbb{C}P^1$ becomes a Riemannian submersion. Then for any unit vector $X \in \mathfrak{su}(2)$ with respect to the Killing form, if $X_P$ is the corresponding left-invariant vector field on $SU(2)$, then the orthogonal projection of $\nabla_{X_P}X_P$ onto $E^\perp = (E+[E,E])^\perp$ is precisely the mean curvature vector field
$$
 H = -\frac{1}{2}\operatorname{grad}\ell \neq 0
$$
of the Riemannian submersion $SU(2) \to \mathbb{C}P^1$. More generally, you'll run into trouble if $E$ is the tangent bundle to a foliation of $Q$ by some locally free isometric action of a connected Lie group, in which case, the obstruction to your claim is the orbit-wise second fundamental form (i.e., O'Neill's $T$-tensor for the Riemannian foliation $E$).
