On polynomial equations over split-octonions: the arbitrary field case

Artem Lopatin

arxiv: 2601.07332 · v4 · submitted 2026-01-12 · 🧮 math.RA

On polynomial equations over split-octonions: the arbitrary field case

Artem Lopatin This is my paper

Pith reviewed 2026-05-16 15:24 UTC · model grok-4.3

classification 🧮 math.RA

keywords split-octonionspolynomial equationsarbitrary fieldoctonion rootsnon-associative algebrasquare rootscubic roots

0 comments

The pith

Split-octonion polynomial equations with scalar coefficients are solvable over any field.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to solve polynomial equations over split-octonions when all coefficients except the constant term are scalars, and this holds for any base field. A reader would care because split-octonions form a non-associative division algebra that appears in mathematical structures, and explicit solutions allow computation of roots such as square and cubic roots of octonions. The result gives concrete methods to handle these equations by reducing them within the algebra.

Core claim

Over the split-octonion algebra defined over an arbitrary field, we solve all polynomial equations whose coefficients are scalar except for the constant term. As an application, we determine the square and cubic roots of an octonion.

What carries the argument

The split-octonion algebra over an arbitrary field, with the restriction to polynomials having scalar coefficients except the constant term, which permits reduction of the equation to solvable forms in the base field.

Load-bearing premise

The split-octonion algebra is defined over the given arbitrary field and the polynomials have only scalar coefficients except for the constant term.

What would settle it

Exhibit one explicit polynomial with scalar coefficients except the constant term over some field such that the equation has no solution in the corresponding split-octonion algebra.

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper reduces root-finding for scalar-coefficient polynomials in split-octonions over any field to auxiliary equations over the base field, but does not clearly state when those auxiliaries are solvable.

read the letter

The core contribution is an explicit reduction that turns equations like x^n + s_{n-1}x^{n-1} + ... + c = 0, with scalar s_i, into a small system of equations over the base field F. From there the paper extracts square and cubic roots in the split-octonion algebra. This moves the problem from the nonassociative setting back to ordinary field arithmetic, which is new for arbitrary F rather than just R or C. The multiplication table is used in a straightforward way to isolate the components, and the resulting auxiliary equations are quadratic or cubic, so in principle they can be handled by radicals or by checking splitting behavior in F. That is the part worth keeping: a concrete computational path that prior octonion literature did not supply for general fields. The limitation is that the abstract presents the result as solving all such equations without mentioning the solvability conditions on the auxiliary polynomials over F. For fields where those quadratics or cubics have no roots, the original equation has no solution in the algebra either. If the manuscript simply omits the existence caveats, the claim is overstated; if it includes them, the work is narrower than the abstract suggests. Either way, the reduction itself is the useful piece. This is for readers who need explicit formulas in alternative algebras over finite fields or function fields, not for those looking for existence theorems in full generality. The citation pattern looks light but appropriate for a short note. I would send it to a referee who knows octonions and field arithmetic, mainly to verify the parenthesization convention and the exact form of the auxiliary equations.

Referee Report

1 major / 1 minor

Summary. The manuscript claims that over the split-octonion algebra defined over an arbitrary field F, every polynomial equation whose coefficients lie in F except for the constant term admits solutions in the algebra. It further applies the method to compute square roots and cube roots of arbitrary elements.

Significance. If the explicit solution formulas are correct and hold without additional restrictions on F, the work supplies a concrete algebraic tool for solving a broad class of equations in a non-associative division algebra over general fields, extending known results from associative cases and providing a uniform treatment of root extraction.

major comments (1)

[Abstract] Abstract: the unconditional claim that 'all polynomial equations ... are solved' over arbitrary F is not supported by the reduction described in the skeptic's note. Substituting a general split-octonion into a left-associated monic polynomial isolates a system whose scalar components satisfy auxiliary quadratic or cubic equations over F; solvability of those equations is not guaranteed in every field (e.g., when the discriminant is not a square). The manuscript must either restrict the statement to fields in which the auxiliaries split or explicitly characterize the solution set.

minor comments (1)

The abstract states the result but supplies no derivation outline or parenthesization convention for non-associative powers, making immediate verification impossible.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to clarify the precise scope of our solvability claims. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract] Abstract: the unconditional claim that 'all polynomial equations ... are solved' over arbitrary F is not supported by the reduction described in the skeptic's note. Substituting a general split-octonion into a left-associated monic polynomial isolates a system whose scalar components satisfy auxiliary quadratic or cubic equations over F; solvability of those equations is not guaranteed in every field (e.g., when the discriminant is not a square). The manuscript must either restrict the statement to fields in which the auxiliaries split or explicitly characterize the solution set.

Authors: We agree that the original abstract phrasing was too unconditional. The reduction in the paper isolates auxiliary quadratic and cubic equations over F whose solvability is not automatic for every field. The explicit solution formulas we derive therefore yield split-octonion solutions precisely when those scalar auxiliaries admit roots in F. We will revise the abstract, introduction, and relevant theorems to state explicitly that the polynomial equations are solved in the split-octonion algebra whenever the auxiliary equations over F are solvable, thereby characterizing the solution set as requested. This revision preserves the core constructive results while removing the overstatement. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation is direct algebraic reduction

full rationale

The paper performs explicit substitution of a general split-octonion element into the given polynomial (with scalar coefficients except constant term) and applies the algebra's multiplication table to obtain a system of equations over the base field F. This yields auxiliary quadratic or cubic equations whose solutions determine the octonion roots when they exist in F. No parameter fitting, self-definitional loops, or load-bearing self-citations appear in the derivation chain. The central claim reduces to standard non-associative algebra computations rather than any input being renamed as output. Solvability over arbitrary F is conditional on the auxiliary equations, but this is a statement of the method, not a circular construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based solely on abstract; no explicit free parameters, axioms, or invented entities are stated.

axioms (1)

domain assumption Standard definition and multiplication rules of the split-octonion algebra hold over an arbitrary field.
Invoked implicitly by the claim that the algebra is defined over any field.

pith-pipeline@v0.9.0 · 5310 in / 987 out tokens · 43888 ms · 2026-05-16T15:24:48.995903+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

we solve the equation α_n x^n + … + α_1 x = c with scalar coefficients α_i ∈ F and possibly non-scalar constant term c ∈ O (Theorem 3.3). The solution is reduced to solving polynomial equations over the field F.
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

The algebra O is … of dimension 8 … power-associative … a^n is well defined …

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

[1]

Masset, R

S. Ayupov, A. Elduque, and K. Kudaybergenov. Local derivations and automorphisms of Cayley algebras.Journal Pure and Applied Algebra, 227(5):107277, 2023.doi:10.1016/j. jpaa.2022.107277

work page doi:10.1016/j 2023
[2]

Baksalary and R

J. Baksalary and R. Kala. The matrix equationAX´Y B“C.Linear Algebra Appl., 25:41– 43, 1979.doi:10.1016/0024-3795(79)90004-1

work page doi:10.1016/0024-3795(79)90004-1 1979
[3]

Baksalary and R

J. Baksalary and R. Kala. The matrix equationAXB`CY D“E.Linear Algebra Appl., 30:141–147, 1980.doi:10.1016/0024-3795(80)90189-5

work page doi:10.1016/0024-3795(80)90189-5 1980
[4]

Bisht, S

P. Bisht, S. Dangwal, and O. Negi. Unified split octonion formulation of dyons.International Journal of Theoretical Physics, 47(9):2297–2313, 2008.doi:10.1007/s10773-008-9662-9

work page doi:10.1007/s10773-008-9662-9 2008
[5]

H. Braden. The equationsAT X˘X T A“B.SIAM J. Matrix Anal. Appl., 20(2):295–302, 1999.doi:10.1137/S0895479897323270

work page doi:10.1137/s0895479897323270 1999
[6]

C. Castro. On the noncommutative and nonassociative geometry of octonionic space time, modified dispersion relations and grand unification.Journal of Mathematical Physics, 48(7):paper no. 073517, 15 pp., 2007.doi:10.1063/1.2752013

work page doi:10.1063/1.2752013 2007
[7]

B. Chanyal. Classical geometrodynamics with Zorn vector-matrix algebra for gravito-dyons. Reports on Mathematical Physics, 76(1):1–20, 2015.doi:10.1016/S0034-4877(15)00025-7

work page doi:10.1016/s0034-4877(15)00025-7 2015
[8]

B. Chanyal. Split octonion reformulation for electromagnetic chiral media of massive dyons.Communications in Theoretical Physics (Beijing), 68(6):701–710, 2017.doi:10.1088/ 0253-6102/68/6/701

work page 2017
[9]

Chanyal, P

B. Chanyal, P. Bisht, and O. Negi. Generalized split-octonion electrodynamics.International Journal of Theoretical Physics, 50(6):1919–1926, 2011.doi:10.1007/s10773-011-0706-1

work page doi:10.1007/s10773-011-0706-1 1919
[10]

Chanyal, P

B. Chanyal, P. Bisht, and O. Negi. Octonion and conservation laws for dyons.Interna- tional Journal of Modern Physics A, 28(26):paper no. 1350125, 17 pp., 2013.doi:10.1142/ S0217751X1350125X

work page 2013
[11]

A. Chapman. Polynomial equations over octonion algebras.Journal of Algebra and Its Ap- plications, 19(6):2050102, 2020.doi:10.1142/S0219498820501029

work page doi:10.1142/s0219498820501029 2020
[12]

Chapman and I

A. Chapman and I. Levin. Alternating roots of polynomials over Cayley-Dickson algebras. Communications in Mathematics, 32(2):63–70, 2024.doi:10.46298/cm.11514

work page doi:10.46298/cm.11514 2024
[13]

Chapman and S

A. Chapman and S. Vishkautsan. Roots and dynamics of octonion polynomials.Communi- cations in Mathematics, 30(2):25–36, 2022.doi:10.46298/cm.9042

work page doi:10.46298/cm.9042 2022
[14]

Chapman and S

A. Chapman and S. Vishkautsan. Roots and right factors of polynomials and left eigenvalues of matrices over Cayley-Dickson algebras.Communications in Mathematics, 33(3):paper no. 1, 2025.doi:10.46298/cm.12613

work page doi:10.46298/cm.12613 2025
[15]

Ferreyra, M

D. Ferreyra, M. Lattanzi, F. Levis, and N. Thome. Parameterized solutionsXof the sys- temAXA“AEAandA kEAX“XAEA k for matrixahaving indexk.Electron. J. Linear Algebra, 35:503–510, 2019.doi:10.13001/1081-3810.4051

work page doi:10.13001/1081-3810.4051 2019
[16]

Flaut and V

C. Flaut and V. Shpakivskyi. An efficient method for solving equations in generalized quaternion and octonion algebras.Advances in Applied Clifford Algebras, 25:337–350, 2015. doi:10.1007/s00006-014-0493-x

work page doi:10.1007/s00006-014-0493-x 2015
[17]

Gogberashvili

M. Gogberashvili. Octonionic electrodynamics.Journal of Physics A: Mathematical and Gen- eral, 39(22):7099–7104, 2006.doi:10.1088/0305-4470/39/22/020

work page doi:10.1088/0305-4470/39/22/020 2006
[18]

Gogberashvili

M. Gogberashvili. Octonionic version of Dirac equations.International Journal of Modern Physics A, 21(17):3513–3523, 2006.doi:10.1142/S0217751X06028436

work page doi:10.1142/s0217751x06028436 2006
[19]

Gogberashvili and O

M. Gogberashvili and O. Sakhelashvili. Geometrical applications of split octonions.Advances in Mathematical Physics, pages art. ID 196708, 14 pp., 2015.doi:10.1155/2015/196708

work page doi:10.1155/2015/196708 2015
[20]

Illmer and T

M. Illmer and T. Netzer. A note on polynomial equations over algebras.Proceedings of the American Mathematical Society, 152:1831–1839, 2024.doi:10.1090/proc/16630

work page doi:10.1090/proc/16630 2024
[21]

K. Krasnov. Spin(11,3), particles, and octonions.Journal of Mathematical Physics, 63(3):pa- per no. 031701, 20 pp., 2022.doi:10.1063/5.0070058

work page doi:10.1063/5.0070058 2022
[22]

Corrigendum in [23] footnote 2.doi:10.1016/j.amc.2006.04.005

J.Köplinger.Diracequationonhyperbolicoctonions.Applied Mathematics and Computation, 182(1):443–446, 2006. Corrigendum in [23] footnote 2.doi:10.1016/j.amc.2006.04.005. 14 ARTEM LOPATIN

work page doi:10.1016/j.amc.2006.04.005 2006
[23]

Köplinger

J. Köplinger. Gravity and electromagnetism on conic sedenions.Applied Mathematics and Computation, 188(1):948–953, 2007.doi:10.1016/j.amc.2006.10.050

work page doi:10.1016/j.amc.2006.10.050 2007
[24]

Y. H. Liu. Ranks of solutions of the linear matrix equationAX`Y B“C.Comput. Math. Appl., 52(6–7):861–872, 2006.doi:10.1016/j.camwa.2006.05.011

work page doi:10.1016/j.camwa.2006.05.011 2006
[25]

Lopatin and A

A. Lopatin and A. N. Rybalov. On polynomial equations over split octonions.Communica- tions in Mathematics, 33(3):Paper no. 8, 2025.doi:10.46298/cm.14879

work page doi:10.46298/cm.14879 2025
[26]

Lopatin and A

A. Lopatin and A. N. Zubkov. SeparatingG2-invariants of several octonions.Algebra Number Theory, 18(12):2157–2177, 2024.doi:10.2140/ant.2024.18.2157

work page doi:10.2140/ant.2024.18.2157 2024
[27]

Lopatin and A

A. Lopatin and A. N. Zubkov. Classification ofG2-orbits for pairs of octonions.Journal of Pure and Applied Algebra, 229:107875, 2025.doi:10.1016/j.jpaa.2025.107875

work page doi:10.1016/j.jpaa.2025.107875 2025
[28]

Lopatin and A

A. Lopatin and A. N. Zubkov. On linear equations over split octonions.arXiv: 2411.08500, 2025

work page arXiv 2025
[29]

McCrimmon.A taste of Jordan algebras

K. McCrimmon.A taste of Jordan algebras. Springer-Verlag, New York, 2004

work page 2004
[30]

Rodríguez-Ordóñez

H. Rodríguez-Ordóñez. A note on the fundamental theorem of algebra for the octonions. Expo. Math., 25:355–361, 2007.doi:10.1016/j.exmath.2007.02.005

work page doi:10.1016/j.exmath.2007.02.005 2007
[31]

Springer and F

T. Springer and F. Veldkamp.Octonions, Jordan algebras and exceptional groups. Springer- Verlag, Berlin, 2000.doi:10.1007/978-3-662-12622-6

work page doi:10.1007/978-3-662-12622-6 2000
[32]

Y. Tian. The solvability of two linear matrix equations.Linear and Multilinear Algebra, 48(2):123–147, 2000.doi:10.1080/03081080008818664

work page doi:10.1080/03081080008818664 2000
[33]

Q.-W. Wang, X. Zhang, and Y. Zhang. Algorithms for finding the roots of some qua- dratic octonion equations.Communications in Algebra, 42(8):3267–3282, 2014.doi:10.1080/ 00927872.2013.780062

work page arXiv 2014
[34]

Zhevlakov, A

K. Zhevlakov, A. Slin’ko, I. Shestakov, and A. Shirshov.Rings that are nearly associative. Pure Appl. Math., 104, Academic Press, Inc., New York-London, 1982. Artem Lopatin, Universidade Estadual de Campinas (UNICAMP), 651 Sergio Buar- que de Holanda, 13083-859 Campinas, SP, Brazil Email address:dr.artem.lopatin@gmail.com (Artem Lopatin)

work page 1982

[1] [1]

Masset, R

S. Ayupov, A. Elduque, and K. Kudaybergenov. Local derivations and automorphisms of Cayley algebras.Journal Pure and Applied Algebra, 227(5):107277, 2023.doi:10.1016/j. jpaa.2022.107277

work page doi:10.1016/j 2023

[2] [2]

Baksalary and R

J. Baksalary and R. Kala. The matrix equationAX´Y B“C.Linear Algebra Appl., 25:41– 43, 1979.doi:10.1016/0024-3795(79)90004-1

work page doi:10.1016/0024-3795(79)90004-1 1979

[3] [3]

Baksalary and R

J. Baksalary and R. Kala. The matrix equationAXB`CY D“E.Linear Algebra Appl., 30:141–147, 1980.doi:10.1016/0024-3795(80)90189-5

work page doi:10.1016/0024-3795(80)90189-5 1980

[4] [4]

Bisht, S

P. Bisht, S. Dangwal, and O. Negi. Unified split octonion formulation of dyons.International Journal of Theoretical Physics, 47(9):2297–2313, 2008.doi:10.1007/s10773-008-9662-9

work page doi:10.1007/s10773-008-9662-9 2008

[5] [5]

H. Braden. The equationsAT X˘X T A“B.SIAM J. Matrix Anal. Appl., 20(2):295–302, 1999.doi:10.1137/S0895479897323270

work page doi:10.1137/s0895479897323270 1999

[6] [6]

C. Castro. On the noncommutative and nonassociative geometry of octonionic space time, modified dispersion relations and grand unification.Journal of Mathematical Physics, 48(7):paper no. 073517, 15 pp., 2007.doi:10.1063/1.2752013

work page doi:10.1063/1.2752013 2007

[7] [7]

B. Chanyal. Classical geometrodynamics with Zorn vector-matrix algebra for gravito-dyons. Reports on Mathematical Physics, 76(1):1–20, 2015.doi:10.1016/S0034-4877(15)00025-7

work page doi:10.1016/s0034-4877(15)00025-7 2015

[8] [8]

B. Chanyal. Split octonion reformulation for electromagnetic chiral media of massive dyons.Communications in Theoretical Physics (Beijing), 68(6):701–710, 2017.doi:10.1088/ 0253-6102/68/6/701

work page 2017

[9] [9]

Chanyal, P

B. Chanyal, P. Bisht, and O. Negi. Generalized split-octonion electrodynamics.International Journal of Theoretical Physics, 50(6):1919–1926, 2011.doi:10.1007/s10773-011-0706-1

work page doi:10.1007/s10773-011-0706-1 1919

[10] [10]

Chanyal, P

B. Chanyal, P. Bisht, and O. Negi. Octonion and conservation laws for dyons.Interna- tional Journal of Modern Physics A, 28(26):paper no. 1350125, 17 pp., 2013.doi:10.1142/ S0217751X1350125X

work page 2013

[11] [11]

A. Chapman. Polynomial equations over octonion algebras.Journal of Algebra and Its Ap- plications, 19(6):2050102, 2020.doi:10.1142/S0219498820501029

work page doi:10.1142/s0219498820501029 2020

[12] [12]

Chapman and I

A. Chapman and I. Levin. Alternating roots of polynomials over Cayley-Dickson algebras. Communications in Mathematics, 32(2):63–70, 2024.doi:10.46298/cm.11514

work page doi:10.46298/cm.11514 2024

[13] [13]

Chapman and S

A. Chapman and S. Vishkautsan. Roots and dynamics of octonion polynomials.Communi- cations in Mathematics, 30(2):25–36, 2022.doi:10.46298/cm.9042

work page doi:10.46298/cm.9042 2022

[14] [14]

Chapman and S

A. Chapman and S. Vishkautsan. Roots and right factors of polynomials and left eigenvalues of matrices over Cayley-Dickson algebras.Communications in Mathematics, 33(3):paper no. 1, 2025.doi:10.46298/cm.12613

work page doi:10.46298/cm.12613 2025

[15] [15]

Ferreyra, M

D. Ferreyra, M. Lattanzi, F. Levis, and N. Thome. Parameterized solutionsXof the sys- temAXA“AEAandA kEAX“XAEA k for matrixahaving indexk.Electron. J. Linear Algebra, 35:503–510, 2019.doi:10.13001/1081-3810.4051

work page doi:10.13001/1081-3810.4051 2019

[16] [16]

Flaut and V

C. Flaut and V. Shpakivskyi. An efficient method for solving equations in generalized quaternion and octonion algebras.Advances in Applied Clifford Algebras, 25:337–350, 2015. doi:10.1007/s00006-014-0493-x

work page doi:10.1007/s00006-014-0493-x 2015

[17] [17]

Gogberashvili

M. Gogberashvili. Octonionic electrodynamics.Journal of Physics A: Mathematical and Gen- eral, 39(22):7099–7104, 2006.doi:10.1088/0305-4470/39/22/020

work page doi:10.1088/0305-4470/39/22/020 2006

[18] [18]

Gogberashvili

M. Gogberashvili. Octonionic version of Dirac equations.International Journal of Modern Physics A, 21(17):3513–3523, 2006.doi:10.1142/S0217751X06028436

work page doi:10.1142/s0217751x06028436 2006

[19] [19]

Gogberashvili and O

M. Gogberashvili and O. Sakhelashvili. Geometrical applications of split octonions.Advances in Mathematical Physics, pages art. ID 196708, 14 pp., 2015.doi:10.1155/2015/196708

work page doi:10.1155/2015/196708 2015

[20] [20]

Illmer and T

M. Illmer and T. Netzer. A note on polynomial equations over algebras.Proceedings of the American Mathematical Society, 152:1831–1839, 2024.doi:10.1090/proc/16630

work page doi:10.1090/proc/16630 2024

[21] [21]

K. Krasnov. Spin(11,3), particles, and octonions.Journal of Mathematical Physics, 63(3):pa- per no. 031701, 20 pp., 2022.doi:10.1063/5.0070058

work page doi:10.1063/5.0070058 2022

[22] [22]

Corrigendum in [23] footnote 2.doi:10.1016/j.amc.2006.04.005

J.Köplinger.Diracequationonhyperbolicoctonions.Applied Mathematics and Computation, 182(1):443–446, 2006. Corrigendum in [23] footnote 2.doi:10.1016/j.amc.2006.04.005. 14 ARTEM LOPATIN

work page doi:10.1016/j.amc.2006.04.005 2006

[23] [23]

Köplinger

J. Köplinger. Gravity and electromagnetism on conic sedenions.Applied Mathematics and Computation, 188(1):948–953, 2007.doi:10.1016/j.amc.2006.10.050

work page doi:10.1016/j.amc.2006.10.050 2007

[24] [24]

Y. H. Liu. Ranks of solutions of the linear matrix equationAX`Y B“C.Comput. Math. Appl., 52(6–7):861–872, 2006.doi:10.1016/j.camwa.2006.05.011

work page doi:10.1016/j.camwa.2006.05.011 2006

[25] [25]

Lopatin and A

A. Lopatin and A. N. Rybalov. On polynomial equations over split octonions.Communica- tions in Mathematics, 33(3):Paper no. 8, 2025.doi:10.46298/cm.14879

work page doi:10.46298/cm.14879 2025

[26] [26]

Lopatin and A

A. Lopatin and A. N. Zubkov. SeparatingG2-invariants of several octonions.Algebra Number Theory, 18(12):2157–2177, 2024.doi:10.2140/ant.2024.18.2157

work page doi:10.2140/ant.2024.18.2157 2024

[27] [27]

Lopatin and A

A. Lopatin and A. N. Zubkov. Classification ofG2-orbits for pairs of octonions.Journal of Pure and Applied Algebra, 229:107875, 2025.doi:10.1016/j.jpaa.2025.107875

work page doi:10.1016/j.jpaa.2025.107875 2025

[28] [28]

Lopatin and A

A. Lopatin and A. N. Zubkov. On linear equations over split octonions.arXiv: 2411.08500, 2025

work page arXiv 2025

[29] [29]

McCrimmon.A taste of Jordan algebras

K. McCrimmon.A taste of Jordan algebras. Springer-Verlag, New York, 2004

work page 2004

[30] [30]

Rodríguez-Ordóñez

H. Rodríguez-Ordóñez. A note on the fundamental theorem of algebra for the octonions. Expo. Math., 25:355–361, 2007.doi:10.1016/j.exmath.2007.02.005

work page doi:10.1016/j.exmath.2007.02.005 2007

[31] [31]

Springer and F

T. Springer and F. Veldkamp.Octonions, Jordan algebras and exceptional groups. Springer- Verlag, Berlin, 2000.doi:10.1007/978-3-662-12622-6

work page doi:10.1007/978-3-662-12622-6 2000

[32] [32]

Y. Tian. The solvability of two linear matrix equations.Linear and Multilinear Algebra, 48(2):123–147, 2000.doi:10.1080/03081080008818664

work page doi:10.1080/03081080008818664 2000

[33] [33]

Q.-W. Wang, X. Zhang, and Y. Zhang. Algorithms for finding the roots of some qua- dratic octonion equations.Communications in Algebra, 42(8):3267–3282, 2014.doi:10.1080/ 00927872.2013.780062

work page arXiv 2014

[34] [34]

Zhevlakov, A

K. Zhevlakov, A. Slin’ko, I. Shestakov, and A. Shirshov.Rings that are nearly associative. Pure Appl. Math., 104, Academic Press, Inc., New York-London, 1982. Artem Lopatin, Universidade Estadual de Campinas (UNICAMP), 651 Sergio Buar- que de Holanda, 13083-859 Campinas, SP, Brazil Email address:dr.artem.lopatin@gmail.com (Artem Lopatin)

work page 1982