Geodesic
The lengths of the quaternion and octotion algebras
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXIX, Tome 453 (2016), pp. 22-32 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {A. E. Guterman and D. K. Kudryavtsev},
     title = {The lengths of the quaternion and octotion algebras},
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A. E. Guterman; D. K. Kudryavtsev. The lengths of the quaternion and octotion algebras. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXIX, Tome 453 (2016), pp. 22-32. http://geodesic.mathdoc.fr/item/ZNSL_2016_453_a2/