  We show that the square of Carnot-Carath\'eodory distance from the origin, in
ideal Carnot groups, enjoys the horizontal semiconcavity (h-semiconcavity)
everywhere in the group including the origin. We apply this property to show
h-semiconcavity for the solutions of a class of non-coercive evolutive
Hamilton-Jacobi equations, by using the associated Hopf-Lax solutions.
