Using the fundamental tensors of a smooth loop and the di_erential geometric characterization of smooth hyporeductive loops, the Akivis operations of a local smooth hyporeductive loop are expressed through the two binary and the one ternary operations of the hyporeductive triple algebra (h.t.a.) associated with the given hyporeductive loop. Those Akivis operations are also given in terms of Lie brackets of a Lie algebra of vector _elds with the hyporeductive decomposition which generalizes the reductive decomposition of Lie algebras. A nontrivial real two-dimensional h.t.a. is presented.