Suppose X and Y are two unit length vector fields on a Riemannian manifold which are orthogonal at each point. Is it true that the lie bracket of X, Y belongs to the span of the vector fields at each point in the manifold?
Is there a way to prove or disprove this without going into local coordinate expressions.
Thanks



Frobenius condition
Pick any manifold of dimension 3 or higher and any two non-vanishing vector fields that do not satisfy the Frobenius condition. Define a Riemannian metric such that the vector fields are orthonormal. 


