Geodesic
Associators and commutators in alternative algebras
Algebra i logika, Tome 58 (2019) no. 4, pp. 479-485

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It is proved that in a unital alternative algebra $A$ of characteristic $\neq 2$, the associator $(a,b,c)$ and the Kleinfeld function $f(a,b,c,d)$ never assume the value $1$ for any elements $a,b,c,d\in A$. Moreover, if $A$ is nonassociative, then no commutator $[a,b]$ can be equal to $1$. As a consequence, there do not exist algebraically closed alternative algebras. The restriction on the characteristic is essential, as exemplified by the Cayley–Dickson algebra over a field of characteristic $2$.
Keywords: alternative algebra, commutator, Kleinfeld function.
Mots-clés : associator 
@article{AL_2019_58_4_a3,
     author = {E. Kleinfeld and I. P. Shestakov},
     title = {Associators and commutators in alternative algebras},
     journal = {Algebra i logika},
     pages = {479--485},
     publisher = {mathdoc},
     volume = {58},
     number = {4},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2019_58_4_a3/}
}
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E. Kleinfeld; I. P. Shestakov. Associators and commutators in alternative algebras. Algebra i logika, Tome 58 (2019) no. 4, pp. 479-485. http://geodesic.mathdoc.fr/item/AL_2019_58_4_a3/