= 1, the left (respectively, right) multiplication Lp (respectively, Rp) by p is an isometry, due to the multiplicativity of the norm. Using [2.2], it follows that the characteristic polynomial of Lp and Rp is x2 − tr(p)x +1)2 and, in particular, the determinant of the multiplication by p is 1, so both Lp and Rp are rotations.
Now, if ψ is a rotation in ℝ4 ≃ ℍ, a = ψ(1) is a norm 1 quaternion, and
so the composition
for any x ∈ ℍ. That is, for any x ∈ ℍ,
It follows that the map
is a surjective (Lie) group homomorphism with ker Ψ = {±(1, 1)}. We thus obtain the isomorphism
and from here we get the isomorphism SO3 (ℝ) × SO3 (ℝ) ≃ PSO4 (ℝ).
Again, this means that it is easy to compose rotations in four-dimensional space, as it reduces to multiplying pairs of norm 1 quaternions: ψp1, q1 ψp2, q2 = ψp1p2, q1q2.
2.2.3. Octonions
In a letter from Graves to Hamilton, dated October 26, 1843, only a few days after the “discovery” of quaternions, Graves writes:
There is still something in the system which gravels me. I have not yet any clear views as to the extent to which we are at liberty arbitrarily to create imaginaries, and to endow them with supernatural properties.
If with your alchemy you can make three pounds of gold, why should you stop there?
Actually, as we have seen in [2.3], the algebra of quaternions is obtained by doubling suitably the field of complex numbers: ℍ = ℂ ⊕ ℂj.
Doubling again we get the octonions (Graves–Cayley):
with multiplication mimicked from [2.2]:
and usual norm: n(p1 + p2l) = n(p1) + n(p2), for p1, p2, q1, q2 ∈ ℍ.
This was already known to Graves, who wrote a letter to Hamilton on December 26, 1843 with his discovery of what he called octaves. Hamilton promised to announce Graves’ discovery to the Irish Royal Academy, but did not do it in time. In 1845, independently, Cayley discovered the octonions and got the credit. Octonions are also called Cayley numbers.
Some properties of this new algebra of octonions are summarized here:
– The norm is multiplicative: n(xy) = n(x)n(y), for any .
– is a division algebra, and it is neither commutative nor associative!
But it is alternative, that is, any two elements generate an associative subalgebra.
A theorem by Zorn (1933) asserts that the only finite-dimensional real alternative division algebras are ℝ, ℂ, ℍ and
– The seven-dimensional Euclidean sphere is not a group (associativity fails), but it constitutes the most important example of a Moufang loop.
– As for ℍ, for any two imaginary octonions u, we have:
for the usual scalar product u ∙ v on
– is again a quadratic algebra: x2 − tr(x)x + n(x)1 = 0 for any , where and , where for x = a1 + u, a ∈ ℝ, , .
And, as it happens for quaternions, octonions are also present in many interesting geometrical situations, here we mention a few:
– the groups Spin7 and Spin8 (universal covers of SO7(ℝ) and SO8(ℝ)) can be described easily in terms of octonions;
– the fact that is a division algebra implies the parallelizability of the seven-dimensional sphere S7. Actually, S1, S3 and S7 are the only parallelizable spheres (Adams 1958; Bott and Milnor 1958; Kervaire 1958);
– the six-dimensional sphere can be identified with the set of norm 1 imaginary units: , and it is endowed with an almost complex structure, inherited from the multiplication of octonions.
S2 and S6 are the only spheres with such structures (Borel and Serre (1953));
– contrary to what happens in higher dimensions, projective planes do not need to be desarguesian. The simplest example of a non-desarguesian projective plane is the octonionic projective plane .
David R. Wilkins has compiled a large amount of material on the work of Hamilton1, and for complete expositions on quaternions and octonions, the interested reader may consult Ebbinghaus et al. (1991) and Conway and Smith (2003).
2.3. Unital composition algebras
Composition algebras constitute a generalization of the classical algebras of the real, complex, quaternion and octonion numbers.
A quadratic form n
