Some remarks about Fibonacci elements in an arbitrary algebra

TL;DR

This paper explores properties of Fibonacci elements within arbitrary algebras, introducing imaginary Fibonacci quaternions and octonions, and establishing their linear independence and product relations.

Contribution

It defines imaginary Fibonacci quaternions and octonions and proves key linear independence and product properties in an arbitrary algebra context.

Findings

Three imaginary Fibonacci quaternions are linearly independent.

The mixed product of three imaginary Fibonacci octonions is zero.

Relations between Fibonacci elements in arbitrary algebras are established.

Abstract

In this paper, we prove some relations between Fibonacci elements in an arbitrary algebra. Moreover, we define imaginary Fibonacci quaternions and imaginary Fibonacci octonions and we prove that always three arbitrary imaginary Fibonacci quaternions are linear independents and the mixed product of three arbitrary imaginary Fibonacci octonions is zero.