On Absolute-Valued Algebras with Nonzero Central Element

Let \(\mathcal {A}\) be an absolute-valued algebra with nonzero element a such that a and \(a^{2}\) are central, then \(\mathcal {A}\) is pre-Hilbert space and admits an involution. We also show that if \(\mathcal {A}\) is an absolute-valued algebra with nonzero central element satisfying \((x,x^{2},x)=(x^{2},x^{2},x^{2})=0\) , then \(\mathcal {A}\) is finite-dimensional, flexible, and isomorphic to either \(\mathbb {R}\) , \(\mathbb {C}\) , \(\stackrel {\star }{\mathbb {C}}\) , \(\mathbb {H}\) , \(\stackrel {\star }{\mathbb {H}}\) , \(\mathbb {O}\) , or \(\stackrel {\star }{\mathbb {O}}\) .