  In this paper, we obtain remainder term representation formulae for the
higher-order Steklov inequality for vector fields which imply short and direct
proofs of the sharp (classical) Steklov inequalities. The obtained results
directly imply sharp Steklov type inequalities for some vector fields
satisfying H\"ormander's condition, for example. We also give representation
formulae for the $L^{2^{m}}$-Friedrichs inequalities for vector fields.
