We study the absolute-valued algebras ${𝒜}$ satisfying the identity $(x{x}^2)x=x(x^{2}x).$ We show that, if moreover, the norm of ${𝒜}$ comes from an inner product and ${𝒜}$ contains a nonzero central element then ${𝒜}$ is finite dimensional and isomorphic to either $ℝ,ℂ$, $\stackrel{\ast }{ℂ}$, $ℍ$, $\stackrel{\ast }{ℍ}$, $𝕆$, or $\stackrel{\ast }{𝕆}$.

中文翻译：

---

关于满足 (xx2)x = x(x2x) 的绝对值代数的说明

我们研究绝对值代数${𝒜}$满足身份$(X{X}^2)X=X(X^{2}X).$我们证明，如果此外，${𝒜}$来自内积和${𝒜}$那么包含一个非零中心元素${𝒜}$是有限维和同构的$ℝ,ℂ$,$\stackrel{*}{ℂ}$,$ℍ$,$\stackrel{*}{ℍ}$,$𝕆$， 或者$\stackrel{*}{𝕆}$.