The following claim can be found in page 27 of Petersen's book "Riemannian Geometry":
"... any three vectors can be extended to vector fields that commute."
I have been unable to prove the statement, mostly because the equations determining the commutativity of vector fields are a system of nonlinear partial differential equations, although their symmetry might make them easier to treat.
I was wondering if someone could help me find an existence theorem for the Cauchy problem of such equations, or figure out a way to prove such a claim.