Split numbers and linear independence of multiplicative inverses

Question

My understanding is that the product of ijk may equal +1 when working with split-quaternions. What are good examples of systems defined such that the product of two (not three) linearly independent (i.e., not jj or kk) elements of unit magnitude may equal +1? I have been poking around http://math.chapman.edu/~jipsen/structures/doku.php/index.html but I have yet to find/recognize the answer to my question via that resource.

Answer 1

You don't actually specify what type of algebraic structure you're considering, so let's figure it out from context:

You ask for linear independence of elements, which implies you have an underlying vector space structure. You then want to take products of those elements, so you have some sort of algebra over tha vector space (possibly non-associative). You reference the element "1" as a unit, so I'll assume your algebra is unital. You want those elements to be of "unit magnitude", so you also have some sort of norm on this algebra.

I suspect the general category you want to be asking about is Clifford algebras. In any case, my example is from the complex numbers, which should be an allowable space no matter what additional requirements come up.

If $z,w$ are two elements from your algebra, and you want $z w = 1$, that means that $z$ and $w$ are units and are inverse of each other. And for any reasonable definition of "magnitude", you should be able to scale your answer so that both of the elements are of unit magnitude.

So to turn your question around, you're asking for an element of your algebra that is linearly independent from its inverse. There are many examples, perhaps the simplest being the primitive cubic roots of unity $z = -1/2 + i\sqrt{3}/2$ and $w = z^2 = -1/2 - i\sqrt{3}/2$. They are linearly independent, of unit magnitude, and inverses.

Edit: Additional requirements have now been added: The system should be a finite dimensional unital normed algebra over the reals (the vector space has to be over a field, so it can't be over the non-negative reals), $z$ and $w$ must have no real part (in other words, ${1, z, w}$ must be linearly independent), and $z$ and $w$ must have $+1$ as a coefficient (with respect to what generating set it is unclear, since they will always have $+1$ as a coefficient in any basis of which they are members).

With these additional requirements, I believe no example exists. Let's take a slightly stronger version of your question and suppose that we're in a composition algebra, which only additionally assumes that the norm comes from a non-degenerate quadratic form (all of your named examples so far are composition algebras). From every such composition algebra, you can define a symmetric bilinear form $\langle v, w \rangle$ that gives rise to the norm, as well as a conjugate $\bar{v} = 2 \langle v, 1 \rangle - v$. In particular, when $v$ "has no real part", that means that $\langle v, 1 \rangle = 0$, and so $\bar{v} = - v$. But we also have from Prop 2.7 on my linked page that when $N(v) = 1$, $v^{-1} = \bar{v}$. Putting these two requirements together gives that $v^{-1} = -v$, and thus, $\{v, v^{-1}\}$ can never be linearly independent. Thus, no such $j$ and $k$ can exist in such an algebra.

So if there are any examples, you would need a finite dimensional unital normed algebra over the reals where the norm doesn't come from a quadratic form, which I think will not look sufficiently close to the "systems" you're considering. (Depending on how you define "norm", there might be none at all.)