# Questions tagged [octonions]

Octonions form a 8-dimensional normed division algebra constructed over the reals. They can be seen as a non-associative (alternative) extension of the quaternions. They have been first defined and studied in the 19th century by John Graves and Arthur Cayley. There are several variants (such as split-octonions) and strong relations with Lie Groups and projective geometry.

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### Is the average associator over a finite subloop of octonions necessarily zero?

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### Automorphism group of formally real Jordan Algebras of hermitian matrices

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### Characteristic polynomial of an $8 \times 8$ symmetric matrix with indeterminate entries related to octonionic multiplication

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### Octonion algebras over number fields [closed]

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### Determinants in Jordan algebras of Euclidean type

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### Unital nonalternative real division algebras of dimension 8

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### When is the determinant an $8$-th power?

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### What finite simple groups we can obtain using octonions?

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### A general form of a maximal totally isotropic subspace in the split octonion algebra

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### Are two octonion algebras with different multiplications isomorphic?

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### A general form of mappings preserving angle between vectors and their image in $R^8$

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### Matrix representation of the automorphisms of the octonion's algebra without Lie's theory

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### About some property of automorphism of octonions

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### Constructing real forms of the Tits-Freudenthal magic square for (Rosenfeld) projective planes

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### Atiyah on the “Galois group of the octonions” and Physics

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### Properties of complexified octonions

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### Generating $\mathfrak{so}(7)$

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### Determinants of octonionic hermitian matrices

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### Exceptional symmetric spaces embedded in exceptional Lie group

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### Transitivity of $Spin(7)$ in triples of vectors

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### Expressing $SO_8$ element as product of $L_u$ and $R_u$ for unit octonions $u$

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### Does the Cayley-Dickson construction preserve isomorphism of quaternion algebras?

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### Looking for Severi varieties

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### Pseudo-holomorphic curves in the six-sphere

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### Is there an octonionic analog of the K3 surface, with implications for stable homotopy groups of spheres?

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### Action of G_2 on certain 7x7 skew-symmetric matrices

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### Cayley Subspaces in a Calibrated 8-Space

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### $Spin(7)$ as stabilizer of a $4$-form revisited

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### Octonions product: inversion in the right and identity in the left

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### $Spin(7)$ as stabilizer of a $4$-form

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### How can the Cayley table for the elements of basis of a Cayley-Dickson algebra be summarized in an algebraic expression?

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### Decomposition of $S^7=Spin(7)/G_2$

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### Grunsky-Motzkin-Schoenberg formula

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### Octonions and the Fano plane.

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### Filling in a rational orthogonal matrix given one row

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### The octonions on a bad day

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