Associators and commutators in alternative algebras

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 $2$.