Number Theory: An Introduction to Mathematics

6 Quaternions and Octonions 53

represented, then−Vis not. On the other hand, supposeuis a pure quaternion, so that
α 0 =0. Thenux+xu=ux+ ̄xu ̄is real, and given by

ux+xu=− 2 (α 1 ξ 1 +α 2 ξ 2 +α 3 ξ 3 )= 2 〈 ̄u,x〉,

with the notation of§10 for inner products inR^3. It follows that

y=uxu ̄= 2 〈 ̄u,x〉 ̄u−x.

But the mappingx→x− 2 〈 ̄u,x〉 ̄uis areflectionin the plane orthogonal to the unit
vectoru. Hence, for every reflectionR,−Ris represented. It may be shown that every
orthogonal transformation ofR^3 is a product of reflections. (Indeed, this is a special
case of a more general result which will beproved in Proposition 17 of Chapter VII.)
It follows that an orthogonal matrixVis represented if and only ifVis a product of
an even number of reflections (or, equivalently, if and only ifVhas determinant 1, as
defined in Chapter V,§1).
Since, by our initial definition of quaternions, the quaternions of norm 1 are just
the 2×2 unitary matrices with determinant 1, our results may be summed up (cf.
Chapter X,§8) by saying that there is a homomorphism of thespecial unitary group
SU 2 (C)onto thespecial orthogonal group S O 3 (R), with kernel{±I}. (Here ‘special’
signifies ‘determinant 1’.)
Since the quaternions of norm 1 may be identified with the points of the unit sphere
S^3 inR^4 it follows that, as a topological space,SO 3 (R)is homeomorphic toS^3 with
antipodal points identified, i.e. to the projective spaceP^3 (R). Similarly (cf. Chapter X,
§8), the topological groupSU 2 (C)is thesimply-connected covering spaceof the topo-
logical groupSO 3 (R).
Again, by considering the mapT:H→Hdefined byTx=vxu−^1 ,whereu,v
are quaternions with norm 1, it may be seen that that there is a homomorphism of the
direct productSU 2 (C)×SU 2 (C)onto the special orthogonal groupSO 4 (R)of 4× 4
real orthogonal matrices with determinant 1, the kernel being{±(I,I)}.
Almost immediately after Hamilton’s invention of quaternions Graves (1844), in a
letter to Hamilton, and Cayley (1845) invented ‘octonions’, also known as ‘octaves’ or
‘Cayley numbers’. We define anoctonionto be an ordered pair(a 1 ,a 2 )of quaternions,
with addition and multiplication defined by

(a 1 ,a 2 )+(b 1 ,b 2 )=(a 1 +b 1 ,a 2 +b 2 ),

(a 1 ,a 2 )·(b 1 ,b 2 )=(a 1 b 1 −b ̄ 2 a 2 ,b 2 a 1 +a 2 b ̄ 1 ).

Then the setOof all octonions is a commutativegroup under addition, i.e. the laws
(A2)–(A5)hold, with 0 =( 0 , 0 )as identity element, and multiplication is both left and
right distributive with respect to addition. The octonion 1 =( 1 , 0 )is a two-sided iden-
tity element for multiplication, and the octonionε=( 0 , 1 )has the propertyε^2 =− 1.
It is easily seen thatα∈Ois in thecentreofO,i.e.αβ=βαfor everyβ∈O,if
and only ifα=(c, 0 )for somec∈R.
Since the mapa→(a, 0 )preserves sums and products, we may regardHas con-
tained inOby identifying the quaternionawith the octonion(a, 0 ). This shows that
multiplication of octonions is in general not commutative. It is also in general not even
associative; for example,