So upon reading the wikipedia entry about the biquaternions I noticed that this algebra has several interesting subalgebras:(adsbygoogle = window.adsbygoogle || []).push({});

1. Thesplit-complexnumbers of the form {σ = x+y()| ∀(x,y)∈ ℝ} which have the norm σ⋅σhi^{*}= (x^{2}-y^{2}).

2. Thetessarineswhich can be written as {α + β| ∀(α,β,)∈ℂj^{1}&= -1}j^{2}

3. Thecoquaternionswhose bases form the dihedral group D4 and are define as the Span{1, i,(hj), (hk)}

But there is a 4th subalgebra that is somewhat similar to the coquaternions. And its elements can be defined as {g= a + b+ ci+ dh|hi∀(a,b,c,d)∈ℝ}. Now of coursei=^{2}h= -1 and^{2}hi=^{2}

h^{2}= (-1)i^{2}^{2}= +1.

But since I'm not sure how to add a grid for the Cayley table I'll also wrote down the other relational equations:

= +(h⋅i)hi

= -(i⋅h)hi

()⋅hi= -ih

⋅(i) = +hi⋅(h

h) = -hii

()⋅hi= +ii

Using these rules it can be shown that { g | g ∈ Span[1,i,h,(hi)]} is closed under products and if we define g^{*}= a - b- ci- dh, then g⋅ghi^{*}= a^{2}+b^{2}+c^{2}-d^{2}= -ds^{2}where ds^{2}is the Minkowski metric.

So does this subalgebra have an official name and could it's elements be used as operators to describe the Lorentz transformation?

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# The biquaternions and friends

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