Associative Composition Algebra/Split-quaternions

There are at least three portals leading to split quaternions: the dihedral group of a square, matrix products in M(2,R), and the modified Cayley-Dickson construction. The work of Max Zorn on split octonions showed the necessity of including split real AC algebras in the aufbau of the category.

The imaginary units v such that v ² = −1 lie on a two-sheeted hyperboloid in split quaternions

The development through the dihedral group was started with a lemma in the Introduction, and is completed with exercises below.

Or one can start with a basis {1, i, j, k} taken from M(2,R), where the identity matrix is one, ${\begin{pmatrix}0&1\\1&0\end{pmatrix}}$ is j, ${\begin{pmatrix}0&1\\-1&0\end{pmatrix}}$ is i, and ${\begin{pmatrix}1&0\\0&-1\end{pmatrix}}$ is k. Some practice with matrix multiplication shows they are anticommutative like division quaternions, but some products differ:

j² = +1 = k²,   j k = − i .

Then the real AC algebra of split-quaternions uses coefficients w, x, y, z ∈ R to express an element, its conjugate, and the quadratic form N:

$q=w+xi+yj+zk,\quad q^{*}=w-xi-yj-zk,\quad N(q)=w^{2}+x^{2}-y^{2}-z^{2}.$

Exercises

1. What are the involutions on a square ?

2. As reflections, what is the angle of incidence of the axes of reflection ?

3. The composition of these reflections has what angle of rotation ?

Computations

Insight into the structure and dynamics of split-quaternions is available through elementary computational exercises. These exercises use j² = +1 = k² and jk = −i, contrary to the quaternion group, which is expressed with the same letters i, j, k, but which here refer to the dihedral group of a square instead.

1. For r = j cos θ + k sin θ, show that r² = +1 = −r r*.
2. Compute ir .
3. Recall that <q, t> = (q t* + t q*)/2. Show <q, t> = real part of q t*
4. Definition: q and t are orthogonal when <q, t> = 0.
5. Show that for any theta, r and ir are orthogonal.
6. Let p = i sinh a + r cosh a. Show that p² = +1 for any a and r.
7. Let v = i cosh a + r sinh a. Show that v² = −1.
8. For a given a and r, show that p and v are orthogonal.
9. Let m = p exp(bp) = sinh b + p cosh b. Show that m m* = −1.
10. Let w = exp(bp) = cosh b + p sinh b. Show that m is orthogonal to w.
11. Show that m is orthogonal to v.
12. For any θ, a, and b defining r, p, w, v, and m, the set {m, w, v, ir} is an orthonormal basis.
13. If u is a unit, <qu, tu> = uu* <q, t>.