# 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.

**13**

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### Why is the automorphism group of the octonions $G_2$ instead of $SO(7)$

**5**

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### Coordinate-free description of an alternating trilinear form on pure octonions

**12**

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### Unit group of octonions over finite fields

**4**

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

**2**

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

**12**

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

**1**

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### Octonion algebras over number fields [closed]

**1**

**2**answers

### Determinants in Jordan algebras of Euclidean type

**6**

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### Unital nonalternative real division algebras of dimension 8

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

**22**

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

**4**

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

**6**

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

**2**

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

**2**

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

**5**

**2**answers

### About some property of automorphism of octonions

**4**

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

**10**

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

**6**

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### Properties of complexified octonions

**1**

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

**11**

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### Determinants of octonionic hermitian matrices

**6**

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### Exceptional symmetric spaces embedded in exceptional Lie group

**3**

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

**8**

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

**6**

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

**5**

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### Looking for Severi varieties

**20**

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

**62**

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

**4**

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

**1**

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### Cayley Subspaces in a Calibrated 8-Space

**2**

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

**3**

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

**8**

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

**4**

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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?

**-1**

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

**6**

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### Grunsky-Motzkin-Schoenberg formula

**19**

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### Octonions and the Fano plane.

**5**

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

**23**

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### The octonions on a bad day

**7**

**3**answers

### Realizing proper pure octonions as conjugates

**46**

**11**answers