Page 209 - Curvature and Homology

EXERCISES 191
7. Let C denote the algebra of Cayby numbers: It has a basis (l,eO,el, .a*, e6)
where I is the unit element and the multiplication table is

the other ei ej being given by permuting the indices cyclically. The algebra C
is non-associative.
Any element of C may be written as
xI+X, XER
where

If x = 0, the element is called a purely imaginary Cayley number. These numbers
form a 7-dimensional subspace E7 C C. The product X . Y of X = E!, xiei E E7
and Y = E:, yie, E E7 may be expressed in the form
X.Y=-(X,Y)I+Xx Y
where

is the scalar product in E7, and

is the vector product of X and Y. The vector product has the properties:
(i) (axl + bX2) x Y = a(Xl x Y) + b(X, x Y),
(i)' X x (cYl + dY,) = c(X x Yl) + d(X x Y,)
for any a, b, c, d E R;
(ii) (X,X x Y) = (Y,X x Y) = 0 and
-
(iii) X x Y = - Y x X.
Consider the unit 6-sphere S6 in E7:
S6 = {X E E7 I (X,X) = 1).
Let g denote the (canonically) induced metric on S6. The tangent space Tx
at X E S6 may be identified with a subspace of E7.