On the generalization of quaternions

Introduction

i²=j²=k²=ijk=ー1

I tried to see if I could extend the quaternions defined by

Bn²＝Cn²＝An・Bn・Cn＝－１

Cn・Bn・An＝ｎ

is assumed to be true.

In conclusion, the quaternions are equivalent to the case n = 1 under this extended system. In that case, A1 = i, B1 = j, and C1 = k are equivalent.

The associative law seems to hold, but I have not been able to prove it.

The table of operations is as follows.

 ×     　An　      　Bn     　 Cn

An   　ーｎ       　Cn  　   ーｎ・Ｂn

Bn   　ｎ/(Cn)  　－１　　ー１/(Ａn)

Cn      Bn             ーAn       －１