  Using the natural irreducible 8-dimensional representation and the two spin
representations of the quantum group $U_q$(D$_4$) of D$_4$, we construct a
quantum analogue of the split octonions and study its properties. We prove that
the quantum octonion algebra satisfies the q-Principle of Local Triality and
has a nondegenerate bilinear form which satisfies a q-version of the
composition property. By its construction, the quantum octonion algebra is a
nonassociative algebra with a Yang-Baxter operator action coming from the
R-matrix of $U_q$(D$_4$). The product in the quantum octonions is a
$U_q$(D$_4$)-module homomorphism. Using that, we prove identities for the
quantum octonions, and as a consequence, obtain at $q = 1$ new ``representation
theory'' proofs for very well-known identities satisfied by the octonions. In
the process of constructing the quantum octonions we introduce an algebra which
is a q-analogue of the 8-dimensional para-Hurwitz algebra.
