Exercise 5, § 19

Suppose that the characteristic of K is $\not= 2$. Let F be a not necessarily associative alternative Cayley K-algebra (III, Appendix, No. 2); we denote its conjugation by $s:x\mapsto \overline{x}$ and its Cayley trace and norm by T and N, respectively. We say that a subalgebra E of F is nondegenerate if for every nonzero element $x$ of E, there exists an $x'\in E$ such that $T(xx')\not= 0$.

a) Prove the equality $x(zy) +z(xy) = T(xz)y= (yx)\overline{z}+ (yz)\overline{x}$ for $x, y, z$ in F (reduce to the case $z=x$).

b) Let E be a subalgebra of F containing 1, distinct from F, stable under $s$, and finite-dimensional over K; suppose that E and F are nondegenerate. Prove that there exists a nonzero element $j$ of F such that $T(xj) = 0$ for every $x\in E$ and $N(j)\not= 0$. We have $\overline{j}=-j,xj=-jx$ for every $x\in E$, and $j^2=\gamma 1_F$ with $\gamma =-N(j)$.

c) Let $x$ and $y$ be elements of E. Prove the formulas $x(yj) = (yx)j= (yj)\overline{x}$ and $(xj)(yj) =\gamma yx$ (use a) and Exercise 2 of III, Appendix, p. 654).

d) Prove that $E + Ej$ is a nondegenerate subalgebra of F, stable under $s$, and of dimension 2 dim(E) that can be identified with the Cayley extension of $(E, s)$ defined by $\gamma$. Deduce from Proposition 3 of III, Appendix, No. 2, p. 614 that E is associative.

e) Deduce from the above that the nondegenerate alternative Cayley K-algebras are the following:

(i) the K-algebra K endowed with the identity involution

(ii) the quadratic algebras over K of type $(\alpha ,0)$ with $\alpha \not= 0$

(iii) the quaternion algebras over K of type $(\alpha ,0, \gamma )$ with $\alpha \gamma \not= 0$

(iv) the octonion algebras over K of type $(\alpha ,0, \gamma , \delta )$ with $\alpha \gamma \delta \not= 0$.

(Construct a sequence of Cayley extensions $K\subset E_1\subset E_2\subset \cdots \subset F$, and observe that an octonion algebra is not associative.)

f ) Prove that an alternative Cayley K-algebra whose nonzero elements are all invertible is one of the aforementioned types.

g) Let R a maximal ordered field. Prove that an alternative Cayley R-algebra whose nonzero elements are all invertible is isomorphic to R, C, H, or O (VIII, p. 368, Remark 1).