The lengths of the quaternion and octotion algebras

The classical Gurvitz theorem claims that there are exactly four normed algebras with division: the real numbers $(\mathbb R)$, complex numbers $(\mathbb C)$, quaternions $(\mathbb H)$, and octonions $(\mathbb O)$. The length of $\mathbb R$ as an algebra over itself is zero; the length of $\mathbb C$ as an $\mathbb R$-algebra equals one. The purpose of the present paper is to prove that the lengths of the $\mathbb R$-algebras of quaternions and octonions equal two and three, respectively.