FOUR-DIMENSIONAL REAL PRE-HILBERT ALGEBRAS SATISFYING ∥x 2∥ = ∥x∥ 2 AND (x, x, x) = 0

Abstract

This paper deals with some results concerning the fourdimensional Pre-Hilbert third-power associative algebras satisfying ∥x
2∥ = ∥x∥
2
. We show that if A contains a central
idempotent e such that ||ex|| = ||x||, then A is flexible and
is isomorphic to H(α) or
∗
H
(α)
where α ̸=
1
2
.