For a given Hurwitz pair $[S\left(Q_S\right),V\left(Q_V\right),o]$ the existence of a bilinear mapping $⭑:C\left(Q_S\right)\times C\left(Q_V\right)\to C\left(Q_V\right)$ (where $C\left(Q_S\right)$ and $C(Q_V$) denote the Clifford algebras of the quadratic forms $Q_S$ and $Q_V$, respectively) generated by the Hurwitz multiplication “o” is proved and the counterpart of the Hurwitz condition on the Clifford algebra level is found. Moreover, a necessary and sufficient condition for "⭑" to be generated by the Hurwitz multiplication is shown.